A deontic premise that leads to a necessity from a permission

Question

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

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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 !}

Answer 1

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.

Answer 2

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.

Answer 3

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.