# A deontic premise that leads to a necessity from a permission

I wanted to devise some rules for myself, then formulate those rules using formal symbolic logic, and one of the rules that I have set for myself is : "Do not say what is unnecessary", in other words : Do not give much information!.

I wrote it down in a formal way to see what implications that statement may have, thus :

For all x, if it is not the case that you ought to tell x, then you ought not tell x" , in Deontic terms :

(∀x)(~OTx → O~Tx)

One of the inferences from that statement is (reverse squiggles) :

(∀x)(P~Tx → ~PTx) , which reads : for all x, if it permissible not to tell x, then it is not permissible to tell x.

That is interesting, let us assume something (let us call it 'a' through instantiation) is permissible to tell a : PTa

What follows from this (using modus tollens) is ~P~Ta (it is not permissible not to tell a).

Which is equivalent to O~~Ta, hence : OTa , You ought to tell a.

I am not sure I quite understand this inference result, my assumption that something is permissible to tell leads directly to the conclusion that I ought to tell it, based on my first rule.

How is that?

(nevertheless, I think it is an interesting, albeit twisted, result).

Note: The reason I think it is a twisted result, is that although I accept the premise (You ought not tell what is unnecessary), I find it difficult to accept that I ought to tell everything that is permissible to tell.

For example : it is permissible to say "cockroach" , but should I say "cockroach" to every stranger I meet?

So, I have the inclination to think that my first premise (rule) must be, somehow, false (i.e : needs some corrections).

Thank you

Symbolic Proof

1. (∀x)(~OTx → O~Tx) {my premise, which I think I agree with}
2. ∴ (∀x)(P~Tx → ~PTx) {from 1, reverse squiggle}
3. ∴ (P~Ta → ~PTa) {from 2, instantiate a}
4. asm : PTa {assume a is permissible to tell}
5. ∴ ~P~Ta {from 3 and 4, modus tollens}
6. ∴ O~~Ta {from 5, reverse squiggle}
7. ∴ OTa {from 6, remove doube not, conclusion : Ought to tell a !}
• "If the situation does not demand telling you ought not to tell", seems agreeable enough, "if the situation does not demand not telling, i.e. if it permits telling, you ought to tell", not so much. But there is a bigger problem. When you mandate not telling you are mandating anything other than telling, including killing the interlocutors, for example. And mandate entails permission. I do not think that was your intention. So you are actually mandating something more restricted. When that is negated in turn, you do not get "tell" under O back. Commented Aug 26, 2019 at 5:31
• @Conifold , Yes, I agree with this "When you mandate not telling you are mandating anything other than telling, including killing the interlocutors, for example. And mandate entails permission" , this is a little bit tricky though ... I will think through this again. Best ! Commented Aug 26, 2019 at 18:30
• I do not think you can do it in SDL, or anything else simple. It forces you to express necessities of different nature as a single O, and take omissions as negations of actions. So when you mandate an omission the double negation rule turns it into mandating action. You can try to disallow the double negation rule, at least under O, or use non-monotone base logic, see Mally. There is a reason deontic logic is plagued with paradoxes, it is ill-conceived as a model of decision-making with priority clashes and free choices. Commented Aug 26, 2019 at 21:29
• It seems like the most immediate fix, but it might mess things up elsewhere. I am not sure what to do with double negations of actions, was even thinking of defaulting them all to T. Commented Aug 27, 2019 at 17:39
• I do not think you should use x in both places, and put it on both sides. If it is ~OTx → O~Tx&OSy (do not tell but say something else, x≠y implicitly), the problem is that mandating not telling is still in the conjunction, saying something else is on top of it. Running it through your argument gives PTx∨P~Sy → OTx, which does not solve the issue. What you want is to restrict what ~Tx can be. But that means distinguishing "elements" of ~Tx (actions), and hanging predicates on them, and in SDL predicates are already reserved for actions themselves. Commented Aug 28, 2019 at 21:51

Your axiom eliminates the distinction between what is permissible and what is obligatory, because it states that what is not obligatory is impermissible. The distinction between obligatory and permissible only exists as long as some things are permissible but not obligatory, but your axiom rules out such cases. Thus you get the result that the obligatory and the permissible are equivalent.

• I see and understand it. But as you may have realized, the formal reformulation comes from a simple premise : "You ought not say it if it is unnecessary". So how can I possibly change that premise to avoid such unexpected implications? Thank you +1 Commented Aug 24, 2019 at 22:06
• @SmootQ I think you need a different necessity operator for 'unnecessary' since presumably it's not deontic necessity ('ought') but rather some sort of conversational necessity. By the way, your rule resembles Grice's maxim of quantity from his theory of conversation.
– E...
Commented Aug 24, 2019 at 22:14
• and can't deontic logic work with a non-deontic necessity? I find it reasonable to think that "ought" and "permissible" and their rules would work fine with any type of necessity or permission? no? Thank you for the reference, I've never heard of Grice's maxim - Best ! Commented Aug 24, 2019 at 22:16
• I would appreciate any example or case that emphasizes the difference between deontic necessity and general necessity where the formal deontic logic fails , thanks ! Commented Aug 24, 2019 at 22:18
• Necessity here means "obligation" not "logical necessity" Commented Aug 24, 2019 at 22:36

I'm thinking you've made something akin to the Four Term Fallacy here. The problem is that you've used the terms 'ought to' and 'is permissible to' as though they are logical synonyms, when in fact (in common usage) they are not. Consider: can there be a case in which it is permissible to say something which ought not to be said? For example, it would be permissible (I imagine) for me to suddenly stop talking about logic and spend a paragraph talking about my car, or my neighbor's cat: nothing wrong with that, exactly. But such a discussion is (ostensibly) superfluous to our discussion of logic, and so I probably ought not to say it.

In one sense you seem to use 'ought' in the rule-oriented sense: that saying statement X violates a rule or precept. In another sense you seem to you 'ought' pragmatically, as a judgement call about whether statement X is useful to the conversation. Trying to make a hard-and-fast rule like "Say nothing except what is necessary" belies the ambiguity of the concept 'necessary', and creates a logical loophole big enough for the proverbial camel.

• Thank you so much for your answer, the problem is that the proof is valid, I verified and there is no formal fallacy like the four term fallacy. I agree though that trying to make a fast rule fails to express any ambiguity that it may contain .. Thanks so much again +1 Commented Aug 24, 2019 at 22:04
• I'd look at it again. The Four Terms fallacy is a sneaky beast, because it plays off of semantic ambiguity. I expect that if you define the term 'ought' explicitly and strictly, then either your curious conclusion will disappear, or you will see it as a function of an impossibly narrow scope. Commented Aug 24, 2019 at 22:24
• I agree, logic is like a sharp razor that can wound you whilst you think you were paying attention , I will add the formal proof to the end of the question . Best ! Commented Aug 24, 2019 at 22:25
• I added the formal proof Commented Aug 24, 2019 at 22:32
• The extra term in a four term fallacy would be independent. But what is obligatory and what is permitted clearly are strongly related. (In particular they are dual aspects of the Cardinal mode.)
– user9166
Commented Aug 29, 2019 at 22:40

You are trying to model a modal situation with non-modal logic. Even as early as Aristotle, we knew this does not work. (Though he notoriously didn't quite get how it did work.) He, and much of the remaining history of logic, focussed on necessity vs possibility in terms of what is dictated by absolute or contingent aspects of reality. But the rules of all modalities follow the same rules of modal logic with different interpretations.

It is clearly not true that whatever is not necessary is necessarily not the case. It is not necessary for me to wear a tie, that does not mean I necessarily won't wear one. Instead, it is only possible that I won't wear one.

This same pattern applies to three distinct moods, each with a pair of boundaries, one with minimal and the other with maximal freedom, corresponding to Aristotle's choice of necessity and possibility.

Aristotle's modes are Fixed modes, they represent rules that literally cannot be violated, but always apply. You are talking about the second mode, the Cardinal mode, which is about rules and expectations that are not fixed in nature but can be violated. And there is a Mutable mode which represents tendencies or decisions under free will which shape our expectations, but to which the concept of violation is simply not relevant. (Unfortunately if you look any of these up you will get eight pages of Astrology or Alchemy references before you get a reference to modal logic.)

We can see these in the modal verbs English inherits from its Germanic roots: can/must, should/may, and would/might (related to German konnen/mussen, sollen/darfen, wollen/mochten) The meanings of these in both languages have shifted from their origins, but they are still fairly close a lot of the time.

What is not necessarily so is possibly not so. What you are not obligated to do, you are permitted not to do. What you are not inclined to do, you are open to not doing. Negation does not travel directly across the modal marker, it 'dualizes' across it, flipping one of the two paired modals into its complement.

So, it is not logical that 'if it is not the case that you ought to tell x, then you ought not tell x'. Negation, as noted above, just does not work that way in a modal context. If it is not the case that you ought to tell x, the most you know is that you may avoid telling x. You have no obligation either way.

Formally ~OTx != O~Tx, ~OTx = P~Tx. (Classically your O would be a box [] (because it is the tighter constraint) and your P would be a diamond <> (because it is the looser constraint). Since this is the second mode they would sometimes have rounded corners, and those for the third mode would often have concave sides. I am not enough of a Unicode maven to look up the symbols.)

Your deduction proceeds from not realizing these two are modals, or actually, are aspects of the same modality.

• I agree with what you said about the trickiness of modal logics, I am aware that '~OTx != O~Tx' .. but remember, the goal is to devise/formulate a deontic statement not to claim that ~OTx is the same as O~Tx . As for modal alethic logic, it has its trickiness too (for instance : necessity works on the whole conditional and not on the antecedent and consequent, example : ◻(A → B) is correct, while (◻A → ◻B ) is incorrect. And I think what's happening with OTx is akin to this kind of anomaly. Thank you so much for your answer +1 ^^ Commented Aug 31, 2019 at 9:37
• I think this is an example of how deontic logic does not work with our day-to-day duties, it's best viewed as a framework for general duties (like categorical imperatives in kantian ethics, moral code...etc). If I understand right. Commented Aug 31, 2019 at 9:42
• I do think that deontic logic works in our day-to-day duties. But aside from the basic structure all modal logic shares, which is pretty weak, it is parallel to law. Law, if you look at it in terms of outcomes, is a paraconsistent logic with statistical truth values -- something rather more complex than most of us want to process. We are better of getting a feel for it than learning the actual content
– user9166
Commented Aug 31, 2019 at 17:56
• I see, yes, or like a fuzzy logic, I will look into these non-classical logics and see if I can find something interesting... thank you so much ! Best Commented Aug 31, 2019 at 22:16