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In logic (or philosophy) jargon, what do you call the generalization of the two mutually exclusive condition states must and must not? I assume mustness isn't the correct term but it conveys what I'm looking for...

Put differently, how do you call the property of being either a "must" or a "must not" condition.

EDIT:

Perhaps it is easier to start and suffice with the logic term for the group containing the mutually exclusive X and not X, or the condition of being/relating to either one of them. I wouldn't call this notness, but you can get the idea of what I'm after...

EDIT 2:

From analyzing the replies it seems the group I had originally meant should have been defined much more rigorously as comprising the following two states:

  1. Must have / be / exist / etc...
  2. Must have not / be not / exist not / etc...

Trying to deconstruct, both the two states are in fact composite structures comprising two elements each: "Must" and a further "Condition".

One state allowed by the group is that condition X must apply, the other allowed group state is that condition X must-not apply. The state that condition X may or may not apply is - not a state allowed by the defined group even though in English saying must not has the interpretation of may or may not / it doesn't matter. That last interpretation is not included in my group definition.

The two types of conditions are mutually exclusive, but the the two aggregate statuses are not different modalities or a bivalence; the opposite of must can also be "does not have to be", so modality/bivalence are probably ambiguous unlike the group definition, or do not apply as a representative description of the group comprising the two (composite) states.

"necessity type" or "necessity requirement" does not capture the relationship between the "Condition" parts, which I could hope there's a word to describe, but at least it does not imply something out of sync with the group's actual definition.

Or, I could call it "condition modality", if "modality" indeed applies. Again, it won't capture the entire specificity of meaning of the group's content, as this group is a modality that allows just two opposite states and not the broad spectrum of modalities, but at least it does not imply something out of sync with the group's definition.

Maybe dichotomous condition modality would capture it all! albeit being a terrible clause to process for most humans, which this question called for anyway ;)

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    This notion is generally called necessity in the case of the "must". Necessarily Q. The other case, necessarily not Q. – virmaior Sep 29 '14 at 3:03
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    With your new edit, you might be looking for bivalence which refers to something that must be either A or not A. – virmaior Sep 30 '14 at 6:46
  • No one has noted outright that your first sentence is just false. 'Must' and 'Must not' are not mutually exclusive states. It is not true that I must pet my dog, or I must not pet my dog. – user9166 Sep 30 '14 at 16:22
  • It is true that I must either pet or not pet my dog. Must, as you are using it just implies the modality of necessity. There is nothing dichotomous or conditional about it. The condition is applied to the mood, in which case there is a dichotomy in the condition, or the mood is applied to the condition, in which case there is no dichotomy at all, and lots of cases outside your frame. The condition is not part of the mood. – user9166 Sep 30 '14 at 17:35
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I think the general notion you are after is 'mood' or 'modality', as it is deployed in modal logic. But English has such bizarre modal structure that we can't tell from the question which pair of must/must not you mean...

In its the most literal meaning, it is the complement of 'can', and the category would be 'necessity', a 'fixed' or 'alethic' mood. In this sense something must happen if it cannot not happen, and it must not happen if it is not true that it might happen or can happen.

But we also use must in a sense that is the complement of 'may', and the category would then be 'obligation', a 'cardinal' or 'deontic' mood. In that sense you must do something if you may-not not do it -- to the degree that I or some set of rules I am referencing have the power of obligation, but not coercion, over you and I say what you may and may not do. In that sense you must not do it if it is emphatically not true that you may.

  • Is there a term going more restricted than the ambiguous English "must"? I am not sure how to grasp the various shades of ambiguity of "must" in English or what does "something must happen if it cannot not happen" imply. Would love to learn... – matt Sep 30 '14 at 10:52
  • In theory there are six moods, from German -- three primary moods: will (German will, implying intent), shall (German soll, implying obligation) and can (German kann, implying possibility); and their complements might (German mag, implying choice), may (German darf, implying permission) and must (German must, implying necessity). But we have let them move around a lot over time, we have let them get confused with our tenses (e.g. will is also the past tense marker, etc.) and we have now we just have to guess a lot. – user9166 Sep 30 '14 at 14:22
  • When I say 'must = cannot not', I mean that, in that sense something 'must' happen if it is not possible for it not to happen -- 'An electron and a proton in empty space must move toward one another'. But we now also use must to mean that it is not permissible for it not to happen, according to some set of rules other than natural law -- 'You must complete these forms. You simply must come see this.'. The word 'may' is even worse -- being used to bestow permission, guess about possibility, or convey intent, with little clue other than inflection to imply which. – user9166 Sep 30 '14 at 14:23
  • The ambiguity of natural language that you're running into is precisely why logicians use formal notations! – digitig Sep 30 '14 at 18:00
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If you allow that e.g. "Come!" has the same connotation as "You must come!" (and allow linguistic jargon as well), you could call it a directive or deontic modality:

Deontic modality is a linguistic modality that indicates how the world ought to be, according to certain norms, expectations, speaker desire, etc.

This category includes the following subcategories:

  • Directive modality (commands, requests, etc.): "Come!", "Let's go!", "You've got to taste this curry!"

EDIT (following OP's EDIT :)

A dichotomy is any splitting of a whole into exactly two non-overlapping parts, meaning it is a procedure in which a whole is divided into two parts. It is a partition of a whole (or a set) into two parts (subsets) that are:

  • jointly exhaustive: everything must belong to one part or the other, and
  • mutually exclusive: nothing can belong simultaneously to both parts.

Such a partition is also frequently called a bipartition [...]

The above applies directly when the term is used in mathematics, philosophy, literature, or linguistics. For example, if there is a concept A, and it is split into parts B and not-B, then the parts form a dichotomy.

  • Not sure what meaning does deontic add on top modality... does it? – matt Sep 30 '14 at 10:45
  • @matt Have you looked at the examples for commissive and volitive modalities too? I can see a difference, although I doubt that these hair-splitting definitions are of much practical use. – Drux Sep 30 '14 at 11:02
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I am taking a gamble here...

Must be x, is that which is true in every possible world. Both Hilary Putnam and Saul Kripke have argued for the existence of a posteriori necessities. Rigidity is a important concept for a posteriori necessities. A term is a rigid designator when it designates the same object in all possible worlds in which that object exists (from Wikipedia) This is an excerpt from William G. Lycan, The Philosophy of language (Routledge)

Kripke offers a further little intuitive test for telling whether a term is rigid: try the term in the sentence frame, 'N might no have been N'. If we plug in, for N, a description like 'the president of the united states in 1970', we obtain 'the president of united states in 1970 might not have been the president of united states in 1970'; and the latter sentence is clearly true, at least on its most natural reading: the person who was President in 1970 might not have been President then (or at any other time). The truth of this sentence shows that the description to refer to different people in different worlds, hence to be flaccid. Buf if we put in the proper name 'Nixon' we get ' Nixon might not have been Nixon' at best a very strange sentence. [...] he could have failed to be named 'Nixon' but that is not to have failed to be Nixon himself (page 47 - 48)

Modal operators are usually written □ for Necessarily and ◇ for Possibly. So the description 'the president of the united states in 1970' must not be true in every possible world. So, ◇ for Possibly (must not be). However, Nixon must be himself in every possible world. So, □ for Necessarily (must be).

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After the edits I think the question is at root a misunderstanding of the way negation works in moods.

You are still after the mood of necessity or obligation, and the basic underlying concept still is mood (or modality, if you are quite modern). But you are thrown by the fact that unlike non-modal concepts like existence, any mood admits two kinds of opposition -- negation and complementarity.

Moods just automatically do that, they have two slots for 'not'. 'Can' is the clearest. It has two opposites -- 'not must' and 'must not' and we actually bother to spell them differently in English 'can not' and 'cannot'.

(The least clear case is 'may', which is a game many English-speaking children play at some point -- 'You may do that' means 'You have permission to do that', but 'You may not do that' can rarely be parsed as 'You have permission not to do that'. It still sounds to many native speakers like it should mean that by default, because that is how our better example of 'can' works. Many a child of middling age has perhaps misunderstood that once, and then tried to get away with the purposeful misinterpretation over and over again. But back to 'can'...)

If something can happen it is not true that it might not happen, and it is true that it must not fail to happen.

Wittgenstein (in The Blue Book) makes great fuzzy hay out of the two different negations of a mood. He labels one of them 'ne' and the other 'non' after some aspect of French grammar.

The 'non' opposite, or direct negation, of 'can' is 'not must' or 'can not'. If it is not true that I must do something, then I can fail or refuse to do it. I can go or I can not go (I can non-go).

The 'ne' opposite, or complement, of 'can' is 'must not' or 'cannot' as everything either can or must not happen, I can go or I cannot go (I un-can go).

Your example of 'must' and 'must not' is a negation linguistically. But it is not a negation in the sense of set-theory. It is not true that for any verb X I either must X or must not X. I could be free -- either I must X or I might not X.

So the truer 'opposite' pair would be the complements 'must' and 'might not' rather than 'must' and its negation 'must not'.

(This is really very simple, but quite hard to keep straight in one's head in English... The language defeats itself.)

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