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The other day, my housemate said "Don't forget to not leave the spoon at the bottom of the container". I understood what he meant: "Do not leave the spoon at the bottom of the container due to forgetfulness".

Turning the words over in my head, I thought of the (equivalent?) expression "Forget to leave the spoon at the bottom of the container" by cancelling the double negation.

However, cancelling the double negation seems to have changed the meaning of the expression in this case. It seems that commanding someone to forget something is distinct from telling them to not forget its negation.

Is there a problem with how I am thinking about this example, or does the verb "forget" just not easily conform to classical first order logical principles? Is it better modeled by a more exotic logic?

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    Imperative statements aren't modelled directly in FOL. You could have look at en.wikipedia.org/wiki/Imperative_logic
    – Fizz
    Apr 14 at 18:09
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    And more like at the "further reading" section of that page, since it only covers the issues with trying to come up with a representation... and stops around 1944. Looking at Gensler's textbook (p. 270) there are indeed problems with "Don’t "... although he only discusses such problem with "don't" covering a conjunction. Here you have an explanation... which is probably even more tricky.
    – Fizz
    Apr 14 at 18:20
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    @Fizz Even dropping the imperative part I still see a difference between "I forgot to not go in to work today" and "I didn't forget to go into work today". Apr 14 at 18:48
  • @Fizz Are there any problems in imperative logic which are not solved by replacing the declarative with the declarative "I think you should ____". All of the examples on the wikipedia page seem to be resolved under this substitution. Apr 14 at 19:20
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    Imperative logic has its own subtleties, but the problem here is not with them. "Forget" is a propositional attitude, it acts on propositions, not objects. This means that you have to model it as an operator applied to propositions of ordinary logic. Hence the original statement is formalized as ~F(~L(S,B)), where L(x,y) is "leave x at y" predicate. You can not move the negation from outside of the F operator to the inside to cancel it there. That this is then put into the imperative mood is a side issue.
    – Conifold
    Apr 14 at 19:36
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The OP question is somewhat complicated by the imperative mood, but I do not believe that the logic of imperatives or deontic logic (see Hansen, Is there a Logic of Imperatives? for an overview) are at fault for bringing about the counterintuitive result. For the purposes of this example we can straightforwardly interpret imperatives as descriptions of a state of affairs with an attached command to bring them about (the Dubislav convention). The problem is with the underlying descriptions.

Think of forgetting to P, substitute "slurp" for P to make it more concrete. Here P is a description and forgetting is a propositional attitude towards this description. Other propositional attitudes, more commonly studied, are knowing (that slurping happened), believing (that slurping happened), etc. Modeling propositional attitudes requires introducing operators into the base logic that act on propositions, in this case we need the forgetting operator F.

What we get is a type of modal logic (in the broad sense). It is still first order, but has additional items in addition to the usual predicate calculus, the operators, see SEP, epistemic logic on how it works for knowledge and belief. By the way, commanding that P can also be conceived as a propositional attitude. The modal logic involved is called deontic logic, but it is tangential here. In the OP example the proposition P is also more complicated than slurping. It involves a two-place predicate L(x,y), "leave x at y", but that does not affect the situation much.

Writing the OP example formally, the original description (that was commanded to happen) was ~F(~L(S,B)). To cancel the double negation we first need to move ~ past F. But the negation ~ does not commute with the forgetting operator F, and we can not move it inside without shifting the meaning. It is quite clear intuitively that not forgetting to slurp and forgetting to not slurp are two different actions. No wonder that F(L(S,B)) is a different description, and commanding it to happen is a different command. Just as "not forgetting to not slurp" and "forgetting to slurp" are two different actions.

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  • I wonder if any non-trivial summary insight we can gain from employing propositional attitude operators towards predicates, given its many restrictions such as you mentioned we subtly cannot do double negation above and the fact that one major difficulty with either propositional attitude or imperative logic is lack of "inference rule"? Apr 15 at 0:06
  • Thanks Conifold! Do you have recommendations for texts which deal with modal logic? Apr 15 at 12:21
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    @StevenGubkin Chellas, Modal Logic is an introductory text that covers deontic logic, Fitting-Mendelsohn, First-Order Modal Logic is harder, but covers modal logics more broadly. I do not know of elementary texts that go into general propositional attitudes, Hegarty, Modality and Propositional Attitudes is pretty demanding.
    – Conifold
    Apr 15 at 17:39
  • @DoubleKnot Attitude operatos often act like quotation marks with the usual referential opacity with respect to manipulations. The general observation that we can not substitute descriptions or quantify into de dicto modal contexts (like most propositional attitudes) played a big historical role in introducing de re modalities by Kripke. On imperatives specifically, Hansen's review I linked is a good survey of obstacles and paradoxes, he even concludes that there are "no argument forms that resemble `imperative inferences'".
    – Conifold
    Apr 15 at 17:55
  • Thx for ur elaboration. I've read some of Hansen's review before my comment yesterday and noticed he mentioned "conditional imperative" problem such as in Poincare's illegal necessity imperative example, thus he advocated a descriptive deontic logic instead. Though imperative logic didn't get logician attention, ironically common people like to use imperative logic much more than predicative functional logic, such as manifested in programming paradigm which is very messy and hard to track down errors... Apr 15 at 20:11
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Modalities in general do not commute with negation. So for (a clear) example, "agent doesn't know that X [is the case]" and "agent knows that not X [is the case]" are obviously different so we would not think to propose that K~x and ~Kx are the same/equivalent.

As a TLDR summary of equivalences for the (relevant part of the) deontic logic vs English:

You must not leave [the spoon at the bottom of the container].
It is mandatory that you do not leave [the spoon at the bottom of the container].
O~L(s,b(c))

You need not leave [the spoon at the bottom of the container].
You don't have to leave [the spoon at the bottom of the container]
It is not mandatory that you leave [the spoon at the bottom of the container].
~OL(s,b(c))

As well known to linguists must and have to (in English) accept positive (non-negated) statements under their scope as well, but "need" (not be confused with "need[s] to") does not, i.e. "John need leave" is ungrammatical but "John must leave" is ok.


The issue with imperatives (unlike modals more generally) is that we don't seem to have much meaning attached to their negation, in contrast to what's negated inside them.

Since a precise (linguistic) mode of "Do not leave ... due to forgetfulness" is apparently not central to the question, we can conceive it as "do not forget the spoon at the bottom of the container" (per Conifold's answer). Furthermore, you can simply think of "do not forget" as "remember to" then.

  • remember to breathe: Rb.
  • remember to add salt to the dish: RA(s,d).
  • remember not to overcook the meat: R~O(m).

But what does it mean to negate an imperative? Surely we can formulate these in a logic just as easily as we can in the "know" (epistemic logic) case, but with the negation of imperatives we seem to have trouble interpeting them; they sound non-sensical as imperatives in natural language, except maybe as pun admonitions:

  • ~Rb: "don't remember to breathe" or "forget to breathe".
  • ~RA(s,d): "don't remember to add salt..." or "forget to add salt..."
  • ~R~O(m): "don't remember not to overcook the meat" or "forget not to overcook the meat)

Note that this is not the same linguistincally as the "forget about" idiom... althout maybe we could read:

  • ~Rb: "forget about breathing"
  • ~RA(s,d): "forget about adding salt to the dish"
  • ~R~O(m): "forget about not overcooking the meat"

These did require some grammatical tweaks though. But the more troublesome part is that the longer sentences, which use predicates under the modality (which linguistically correspond to transitive verbs), already seem to have somewhat unclear meaning, even in this interpretation. Or at least they sound awkward to me. So, this is why I was hesitant about interpreting "don't leave ... due to forgetfulness" just as "forget about ...".


Likewise one can interpret "don't forget" deontically as "impermissible (forbidden, prohibited)". If we take the "ought to be the case" / "it is obligatory that" as the basic deontic modal primitive (aka deontic necessity denoted by O) as it's commonly done in "standard" deontic logic, then impermissible is defined as Ix = O~x. (And permissible as Px = ~O~x = ~Ix.)

However ~Ox is read as "it is omissible that". Which is fairly award and seldom heard in natural language. (Basically it's the same semantic weirdness as [externally] negating imperatives.) SEP (consequently) also notes that

Deontic non-necessity [...] is seldom labeled [as separate operator].

It's more natural to hear "it is optional that", but that one is actually is (~Ox /\ Px), i.e. "omissible and permissible". In a nutshell:

enter image description here

Frankly, I have serious doubts that if you pick untrained people they would make that kind of semantic distinction between "optional" and "omissible" as mandated in deontic logic. (Although I've done a bit of searching on this, there don't seem be any empirical studies on this particular issue, unlike many other in the logic-psychology interface.)

Anyway, since you mentioned first-order logic... there is actually something to be said here about the general relationship between modal operators as an analogy to models of quantification in [many sorted] first-order logic on non-empty domains (i.e. Aristotelian existential import). Best illustrated in the diagram below:

enter image description here

As noted in SEP this observation basically gives the Kripke semantics of modal logics (in general). Formulated in terms of deontic aspects, using serial frames (i.e. if something is obligatory then it is permissible too), that's

Thus, p is obligatory iff it holds in all the i-acceptable worlds, permissible iff it holds in some such world, impermissible iff it holds in no such world, omissible iff its negation holds in some such world, optional iff p holds in some such world, and so does ¬p, and non-optional when p either holds at all such worlds or at none. If a formula is true at every world in any such model of serially-related worlds, then the formula is valid.

Interestingly enough, in Chinese it seems one has almost the exact formulation of this distinction from a modal logic:

Exemption vs. Prohibition (¬□/□¬)

Generally speaking, the insertion of negation confers a specific pragmatic meaning to a proposition, in some cases producing a shift from the propositional to the illocutionary level, thus generating a speech act. More specifically – as observed in deontic logic (Von Wright 1963: 136ff) and in Chinese linguistics investigation (Li Jinxi 1924) – once a normative statement is turned to the negative form, it produces two antithetic and irreducible sentences, either Prohibition (‘it is necessary not to’) or Exemption (‘it is not necessary to’), either bùkě (‘it is not allowed [possible] to’) or kěbù (‘it is allowed [possible] not to’) (Li 1998–1924: 104–105).

The rest of the paper is much less clear to me (as both as non-linguist and not a speaker of Chinese)... but it seems to argue that some Chinese expressions don't actually change their meaning by this kind of swapping, i.e. they have a preferential reading... and there are "suppletion strategies" to actually give them a different reading (i.e. one that would reflect true swapping of the negation position at logical level.) So it looks like all/most natural languages are messy enough that we can't always find the neat reflection of (deontic) logic we might hope for.

The (roughly) equivalent situation in English seems to be that some modals lexically swap position with negation:

John does not have to leave.

means

It is not mandatory for John to leave.

I.e. the lexical order is the same as the logical/semantic order.

In contrast:

John may not leave.

means

It is not permissible for John to leave.

Here the lexical order in "may not" is actually the opposite of the logical/semantic order (in the longer sentence with the same meaning).

Also, if you search the web for "it is omissible that" in quotes it seems you find nothing but (deontic) logic pages. The more natural English language expressions seem to be "it is not mandatory to" or "it is not required to".

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The two negatives are negating entirely different things and so it’s incorrect to think that you can cancel them without changing the meaning of the sentence.

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