Why ought people be rational?

[egg3000]

I have constructed an argument defending the following proposition: people ought to act rationally. The purpose of this post is really to test it; please read it carefully, and be as critical as possible, for I should like to improve it if necessary and, in the event that any particularly devastating assessments are made, do away with it altogether. I also acknowledge that the following is in fact a moral argument, but I have nevertheless decided to post it in the Logic and Philosophy of Mathematics forum as it contains some modal logic.

Firstly, we should define what is meant here by the word "rational". We say that a person is rational if and only if:

a.That person holds a proposition to be true if and only if the falsity of that proposition implies a logical contradiction, and

b. That person holds an argument to be true if and only if the falsity of that argument's conclusion together with the truth of that argument's premises implies a logical contradiction.

So, what we are really saying when we call a person rational is that they only believe certain propositions to be true and others false under and according to particular conditions (i.e. they only believe things to be true if they are logically necessary). When a person believes something to be true because its falsity would imply a logical contradiction, we can say that they have engaged in a rational act. When a person believes a proposition to be true even though its falsity would not imply a logical contradiction, or even though its truth would imply a logical contradiction, we can say that they have engaged in an irrational act.

Roughly then, an irrational act would be:

c. A valuation of a proposition as being true when its falsity does not imply a logic contradiction, and therefore is not true in very model, or a valuation of a proposition as being true in spite of the fact that it implies a logical contradiction within the same model, or

d. The act of holding an argument to be valid when the truth of its premises do not contradict the falsity of its conclusion, or holding an argument to be invalid when the truth of its premises do contradict the falsity of its conclusion. We can make this more simple by taking an argument and converting it into a proposition (for example, some argument 'A ⊢ B' can be converted into the proposition 'A → B'). This way, we can just say that it is an irrational act to hold an argument to be true when its corresponding conditional is not a tautology, and we can say that it is an irrational act to hold an argument to be false when its corresponding conditional is a tautology.

Formally, we can "hold" a proposition to be true or false by giving it the appropriate truth valuation in the relevant model.

One more thing before we address the main question though: it is necessary to clarify a little bit of notation. I will be using modal logic for this argument. Since it is a moral argument, I will translate the '□' operator as "it is morally necessary that" or "it ought to be the case that", and I will translate the '◊' operator as meaning "it is permissible that". Think of it like this: instead of something being necessary in the ordinary sense, we say that it is morally necessary and always happens in a morally perfect world; instead of something being possible in the ordinary sense, we say that it is morally possible or morally permissible. The '~' operator has its usual meaning as 'it is not the case that'. Finally, let 'P' be the proposition "people act rationally".

Now for the main question: why should we act rationally?

Well, suppose there exists some argument from 'a', which is valid and sound, and whose conclusion is the proposition "it is not the case that people should act rationally". (i.e. there exists some valid and sound argument 'a ⊢ ~□P'). It would follow from this supposition that it is not the case that people should act rationally, so:

1. ~□P, world w

Since since '~□P' is definitionally equivalent to '◊~P', it follows that:

2. ◊~P, world w

According to (2), it is permissible not to act rationally. This means that there is some world x where ~P is true:

3. ~P, world x

In other words, we may hold an argument to be false (at least in world x) even when it is sound and valid. But we have supposed that 'a ⊢ ~□P' is such an argument. We can therefore valuate its corresponding conditional, 'a →~□P' as being false at world x within our model. For the conditional 'a →~□P' to be false, 'a' must be true and '~□P' must be false. The falsity of '~□P' is definitionally equivalent to the truth of '□P'. It follows that at world x:

4. □P, world x
5. a, world x

Now, we have supposed that the argument 'a ⊢ ~□P' is valid, which means that its corresponding conditional is a tautology. This means that the proposition 'a →~□P' must be true in all possible worlds. Consequently, 'a →~□P' must be true at world x. Now, for 'a →~□P' to be true, either 'a' must be false or '~□P' must be true. It can't be the case that 'a' is false at world x because, according to (5), 'a' is true at world x. The only option left is that '~□P' is true at world x:

6. ~□P, world x

Since '~□P' is definitionally equivalent to '◊~P', it follows that:

7. ◊~P, world x

According to (7), there is some world accessible from x in which '~P' is true, so:

8. ~P, world y

However, according to (4), '□P' is also true at world x. This means that 'P' is true in every world accessible from world x. Since world y is accessible from world x, it follows that:

9. P, world y

(9) contradicts (8). It follows that there can exist no argument which is valid and sound, and whose conclusion is the proposition "it is not the case that people should act rationally". In short, it is impossible to rationally deny that people ought to act rationally.

Now, a few remarks and I'll be done. Firstly, I understand the argument may seem convoluted, and I apologise for that. However, it is not without reason that I have made it this way: the correct modal system for moral or deontological logic is, if I am not mistaken, still a matter of some debate. I chose, therefore, to construct my argument so that it would work even in system K (whose accessibility relation has no restrictions), thereby ensuring that it should work no matter what modal system you think appropriate for moral reasoning. Secondly, I am aware of the fact that deductive reasoning is not the only way to reason; inductive reasoning is the obvious counterexample. If we wanted to, we could of course just add a clause to our definition of rationality, stating that a person is only rational if, for practical reasons, they also assume the truth of certain propositions when there is a strong probability of those propositions being true (or something to that effect). However, it is only deductive reasoning with which I am concerned in the argument I put forth, so I would much rather put inductive reasoning aside for now, just for the sake of simplicity.

-egg3000

[GreatandWiseTrixie]

I don't see this as proof as why anyone "ought" to act any kind of way. People hold on to delusion because it feels good, because they are taught what good is and that they should feel good and not bad. As well as their own instincts too, people like quick and easy. Also, if your aim is to get delusional people to think clearly, I doubt any of them would even bother to read that long chain of circular logic you put forth. That's some real logic.

[egg3000]

Hi GreatandWiseTrixie, thank you for your reply,

People hold on to delusion because it feels good, because they are taught what good is and that they should feel good and not bad. As well as their own instincts too, people like quick and easy.

I understand that this must be an intuitive position to take, as I've seen other people take the same position commonly enough. However, I feel it necessary to point out that you seem to provide very little argument in support of your claims. Rather, you appear to assert them mostly as though they are a priori truths. Take, for example, the notion that people "are taught what good is and that they should feel good and not bad" - is this claim really true? How do you know that it is? It is surely not a logical truth, for you can negate it without contradiction.

Moreover, I don't think anything you have said really invalidates anything I have said. Firstly, I can say everything I said in my first post and at the same time assert, without contradiction, that people hold onto delusions in order to feel good, that they are taught what goodness is, that they are taught that they should feel good rather than feel bad, and that people generally prefer a quick and easy option over a difficult one.

Furthermore, the aim of this post is not to get delusional people to think clearly; as I stated in the opening paragraph of the post, my aim is to have my argument criticised by others. Once I am sure it is a solid argument, then I may put some effort into simplifying it, but if I were to do that I don't think I would see any use in posting it on this forum, nor do I think the question of whether I would or wouldn't especially relevant.

Finally, I'm not sure how my argument is circular. All I have done is to suppose, for the sake of argument, that there exists a rational justification for irrationality, and then demonstrate that such a supposition leads to a logical contradiction. This means that a rational justification for irrationality does not and can not exist. This is not a circular argument, it's a proof by contradiction.

-egg3000

[GreatandWiseTrixie]

Hi GreatandWiseTrixie, thank you for your reply,

People hold on to delusion because it feels good, because they are taught what good is and that they should feel good and not bad. As well as their own instincts too, people like quick and easy.

I understand that this must be an intuitive position to take, as I've seen other people take the same position commonly enough. However, I feel it necessary to point out that you seem to provide very little argument in support of your claims. Rather, you appear to assert them mostly as though they are a priori truths. Take, for example, the notion that people "are taught what good is and that they should feel good and not bad" - is this claim really true? How do you know that it is? It is surely not a logical truth, for you can negate it without contradiction.

Moreover, I don't think anything you have said really invalidates anything I have said. Firstly, I can say everything I said in my first post and at the same time assert, without contradiction, that people hold onto delusions in order to feel good, that they are taught what goodness is, that they are taught that they should feel good rather than feel bad, and that people generally prefer a quick and easy option over a difficult one.

Furthermore, the aim of this post is not to get delusional people to think clearly; as I stated in the opening paragraph of the post, my aim is to have my argument criticised by others. Once I am sure it is a solid argument, then I may put some effort into simplifying it, but if I were to do that I don't think I would see any use in posting it on this forum, nor do I think the question of whether I would or wouldn't especially relevant.

Finally, I'm not sure how my argument is circular. All I have done is to suppose, for the sake of argument, that there exists a rational justification for irrationality, and then demonstrate that such a supposition leads to a logical contradiction. This means that a rational justification for irrationality does not and can not exist. This is not a circular argument, it's a proof by contradiction.

-egg3000

It's circular, because your topic says "Why people ought to be rational?" And you prove it by saying, being logical, is the most logical choice. Of course being logical is the logical choice. That's why it's called being irrational in the first place. There is a rational justification for irrationality, but you won't find it through this sort of logic. That's the point of being irrational, to avoid logic and hard thought. Logically, this may conserve energy. And benefit selfish individuals...for example the logical choice would be to sacrifice oneself for a group, but their irrationality gets in the way.

[egg3000]

It's circular, because your topic says "Why people ought to be rational?" And you prove it by saying, being logical, is the most logical choice.

No, what I have said is that any rational justification for the normative statement "irrationality is morally permissible" leads to a logical contradiction. However, if any such justification leads to a logical contradiction, it cannot be rational-it follows that no rational justification for that normative statement can even exist. I'm not just saying that being rational is logically the best choice. Rather, I'm saying that if you consider it logically, you will see that it is impossible to provide a coherent justification for irrationality-for as soon as you provide a coherent justification for irrationality, your justification will at once descend into incoherence.

No, if you consider it closely, you will see that the conclusion of my argument is at no point contained in any of the premises of my argument.

There is a rational justification for irrationality, but you won't find it through this sort of logic.

No again, I've just demonstrated that if you use rationality as a means to justifying irrationality, then you undermine the very grounds of your justification. Otherwise, I don't know what other sort of logic you could be referring to, could you describe it for me?

That's the point of being irrational, to avoid logic and hard thought. Logically, this may conserve energy. And benefit selfish individuals...for example the logical choice would be to sacrifice oneself for a group, but their irrationality gets in the way.

All you have done here is to produce a number of particular examples of attempted justifications for irrationality, you have not explained how it is that these examples are exempt from my argument, nor have you explained how my argument is invalid (that is, you haven't demonstrated that the falsity of the conclusion of my argument would not contradict the truth of its premises).

[Impenitent]

deductive reasoning deals only with definitions...

Ludwig had the same problem

-Imp

[GreatandWiseTrixie]

It's circular, because your topic says "Why people ought to be rational?" And you prove it by saying, being logical, is the most logical choice.

No, what I have said is that any rational justification for the normative statement "irrationality is morally permissible" leads to a logical contradiction. However, if any such justification leads to a logical contradiction, it cannot be rational-it follows that no rational justification for that normative statement can even exist. I'm not just saying that being rational is logically the best choice. Rather, I'm saying that if you consider it logically, you will see that it is impossible to provide a coherent justification for irrationality-for as soon as you provide a coherent justification for irrationality, your justification will at once descend into incoherence.

No, if you consider it closely, you will see that the conclusion of my argument is at no point contained in any of the premises of my argument.

There is a rational justification for irrationality, but you won't find it through this sort of logic.

No again, I've just demonstrated that if you use rationality as a means to justifying irrationality, then you undermine the very grounds of your justification. Otherwise, I don't know what other sort of logic you could be referring to, could you describe it for me?

That's the point of being irrational, to avoid logic and hard thought. Logically, this may conserve energy. And benefit selfish individuals...for example the logical choice would be to sacrifice oneself for a group, but their irrationality gets in the way.

All you have done here is to produce a number of particular examples of attempted justifications for irrationality, you have not explained how it is that these examples are exempt from my argument, nor have you explained how my argument is invalid (that is, you haven't demonstrated that the falsity of the conclusion of my argument would not contradict the truth of its premises).

Therein lies your problem. You used the term "morally permissible" morals are like the tide, always a'changing. How can you base a logical, mathematical statement on such a shaky foundation? And your argument is not contradictory, but logically true, based on it's own definitions. Still circular though.

[egg3000]

Therein lies your problem. You used the term "morally permissible" morals are like the tide, always a'changing. How can you base a logical, mathematical statement on such a shaky foundation? And your argument is not contradictory, but logically true, based on it's own definitions. Still circular though.

I haven't based a logical statement on a moral statement, I've done the converse: I've based a moral statement on a logical statement. I'm not, for example, saying that such and such a theorem is true because it would be immoral for it to be false. Rather, I'm saying that such and such a moral principle is true, because a logical contradiction would ensue if it were false.

You may say that morality cannot be grounded in logic, for morality is forever in flux whereas logic is constant. However, I could just as well assert the contrapositive: morality cannot be forever in flux, for it is grounded in logic and logic is constant. Indeed, this is precisely the point I have attempted to demonstrate in my original post.

Furthermore, an argument does not have to be contradictory in order to be invalid. In fact, it's much easier to show an argument to be invalid; all you have to do is show that the truth of its premises and the falsity of its conclusion do not, together, imply a logical contradiction.

Lastly, the mere fact that the logical truth of an argument depends on a set of definitions does not mean that it is circular. If it were the case that the truth of an argument depended on a set of definitions, but that this set of definitions in turn depended on the truth of the argument, then it would be circular-but that is not the case here.

[Wyman]

I think it is circular, given that "'p is false' leads to a contradiction" is equivalent to 'p is true.' Isn't the reductio ad absurdum argument form universal? I.e. Any false statement in a consistent system can be proven to lead to a contradiction.

But, what if we are dealing with an inconsistent or incomplete system?:

If it is possible for p to be true while 'p is false' does not lead to a contradiction, then you would be saying that it is rational for someone to not believe a true proposition. That's problematic.

[egg3000]

I think it is circular, given that "'p is false' leads to a contradiction" is equivalent to 'p is true.'

I don't think the logical equivalence of the two statements makes it circular. Take for example the following proof:

1. P→((P→P)→P) [axiom 1]
2. P→(P→P) [axiom 1]
3. (P→((P→P)→P))→((P→(P→P))→(P→P)) [axiom 2]
4. (P→(P→P))→(P→P) [1,3,MP]
5. (P→P) [2,4,MP]

Each of the propositions in lines 1-4 are logically equivalent to the proposition 'P→P', but this is nevertheless a legitimate proof (within this system). I doubt, for example, that anyone would call a proof of '¬(P∧¬Q) ⊨ (P→Q)' circular, and yet '¬(P∧¬Q)' is logically equivalent to '(P→Q)'.

The mere fact that the result of my argument flows trivially from a set of definitions is not, I think, sufficient warrant for branding it as circular.

Isn't the reductio ad absurdum argument form universal? I.e. Any false statement in a consistent system can be proven to lead to a contradiction.

I'm sorry, I singled this out because I'm missing your point here (sincerely). Could you please elaborate slightly?

But, what if we are dealing with an inconsistent or incomplete system?:

If it is possible for p to be true while 'p is false' does not lead to a contradiction, then you would be saying that it is rational for someone to not believe a true proposition. That's problematic.

Now that, certainly, is a little problematic. I can't really see any way around these cases other than simply biting the bullet and saying that such systems do not accurately model human rationality, though I'm not as yet sure if I feel especially inclined to do so.

[Wyman]

What I believe you are saying is: a person is rational iff they follow the rule of non-contradiction. To follow the rule of non-contradiction is to be logical. So, your premise is that to be rational is to be logical.

But, you perform a slight of hand:

2. ◊~P, world w

According to (2), it is permissible not to act rationally. This means that there is some world x where ~P is true:

3. ~P, world x

And similarly elsewhere. You don't give conditions to convert your modal operators to non-modal. It does not make sense to say from 'it is morally permissible not to act rationally', it follows that 'there is a world where someone acts irrationally.'

If the pope permits me to eat meat on Friday, it does not follow that I eat meat on Friday or that there is a world in which someone eats meat on Friday. As well, if he permits me not to eat meat on Friday, I can eat meat or not. There is no logical connection between your modal operator and the non-modal sentence 'someone acts rationally.'

[egg3000]

you perform a slight of hand:

2. ◊~P, world w

According to (2), it is permissible not to act rationally. This means that there is some world x where ~P is true:

3. ~P, world x

And similarly elsewhere. You don't give conditions to convert your modal operators to non-modal. It does not make sense to say from 'it is morally permissible not to act rationally', it follows that 'there is a world where someone acts irrationally.'

If the pope permits me to eat meat on Friday, it does not follow that I eat meat on Friday or that there is a world in which someone eats meat on Friday. As well, if he permits me not to eat meat on Friday, I can eat meat or not. There is no logical connection between your modal operator and the non-modal sentence 'someone acts rationally.'

Our primitive vocabulary is as follows:

a.) Any uppercase sentence letters from the English Alphabet (e.g. A, B, C, ... ). If we run out of letters, we can just add subscripts to the letters. So subscripted uppercase sentence letters from the English Alphabet are also included in our vocabulary.
b.) The following connectives: ~, →, □, ◊
c.) Parentheses: (, )

Well-formed formulas are defined as follows:

d.) All uppercase sentence letters and uppercase sentence letters with subscripts are well-formed formulas.
e.) The results of replacing a and b with well-formed formulas in '~a', 'a→b', '□a', and '◊a' are also well-formed formulas.
f.) Nothing else is a well-formed formula.

Ordinarily, we simply translate propositions of the form □a or ◊a as "it is necessary that a" or "it is possible that a" respectively. However, we are dealing with deontic logic, and so it is appropriate, rather, to translate such propositions as "it is morally necessary that a" and "it is morally permissible that a".

We say that a well-formed formula is simple if it does not contain any connectives; we say that a well-formed formula is complex otherwise.

Models

A model for Modal Propositional Logic (MPL) consists of: a set W of possible worlds; a two place function f which, to each simple well-formed formula in each possible world, assigns either a value of 1 or 0; and a binary relation R over the set W of possible worlds. We say that some world x is accessible from some world w if and only if Rwx (call R the accessibility relation).

A common and useful way to think of the accessibility relation is like this: if world x is accessible from world w then, from the point of view of the inhabitants of world w, the happenings in world x are possible. However, my argument is a moral argument, and therefore requires us to use deontic logic. Since deontic logic is the logic of normative propositions, we will think of the accessibility relation in the following manner instead: if world x is accessible from world w then, for the inhabitants of world w, all of the happenings in world x are permissible.

So (continuing with your pope analogy), if you're an inhabitant of some world w and the pope says that it is permissible to eat meat on Friday, that does not mean that there is someone in world w who eats meat on Friday. Rather, it means that the inhabitants of world w can conceive of a morally perfect world x where people eat meat on Friday (that is, of course, assuming that what the pope permits is in fact permissible).

Valuations

Let a be any simple well-formed formula, let β and γ be any well-formed formulas in general, let w be any member (possible world) of W, let M be an MPL model, and let V be the valuation function for that model. V is a two-place function defined according to these restrictions:

g.) V(a, w) = f(a, w)
h.) V(~β, w)=1 if and only if V(β, w)=0
i.) V(β→γ, w)=1 if and only if: either V(β, w)=0 or V(γ, w)=1
j.) V(□β, w)=1 if and only if: if Rwx then V(β, x)=1
k.) V(◊β, w)=1 if and only if: there is some x in W such that Rwx and V(β, x)=1

Given that the proposition P is translated as "people act rationally", and given that ◊~P, clauses (g) and (k) make it valid to infer the proposition "there is a world where someone acts irrationally" from the proposition "it is morally permissible not to act rationally". No sleight of hand.

I apologise for all of this, I did want to avoid going into the finer details in order to make my argument more accessible (forgive me, but I couldn't resist that).

[Wyman]

A common and useful way to think of the accessibility relation is like this: if world x is accessible from world w then, from the point of view of the inhabitants of world w, the happenings in world x are possible. However, my argument is a moral argument, and therefore requires us to use deontic logic. Since deontic logic is the logic of normative propositions, we will think of the accessibility relation in the following manner instead: if world x is accessible from world w then, for the inhabitants of world w, all of the happenings in world x are permissible.

'Possible' and 'Permissible' are not the same as true in fact, which is what you are asserting when you assert P. This is what allows you to eventually get from 'People who are logical are rational' to 'People are irrational who believe that people ought to be illogical.'

So (continuing with your pope analogy), if you're an inhabitant of some world w and the pope says that it is permissible to eat meat on Friday, that does not mean that there is someone in world w who eats meat on Friday. Rather, it means that the inhabitants of world w can conceive of a morally perfect world x where people eat meat on Friday

But you don't say 'the inhabitants of world w can conceive of a morally perfect world where P' - you say 'P'; full stop. If those two statements are equivalent, then deontic logic has got some serious problems.

[egg3000]

A common and useful way to think of the accessibility relation is like this: if world x is accessible from world w then, from the point of view of the inhabitants of world w, the happenings in world x are possible. However, my argument is a moral argument, and therefore requires us to use deontic logic. Since deontic logic is the logic of normative propositions, we will think of the accessibility relation in the following manner instead: if world x is accessible from world w then, for the inhabitants of world w, all of the happenings in world x are permissible.

'Possible' and 'Permissible' are not the same as true in fact, which is what you are asserting when you assert P. This is what allows you to eventually get from 'People who are logical are rational' to 'People are irrational who believe that people ought to be illogical.'

So (continuing with your pope analogy), if you're an inhabitant of some world w and the pope says that it is permissible to eat meat on Friday, that does not mean that there is someone in world w who eats meat on Friday. Rather, it means that the inhabitants of world w can conceive of a morally perfect world x where people eat meat on Friday

But you don't say 'the inhabitants of world w can conceive of a morally perfect world where P' - you say 'P'; full stop. If those two statements are equivalent, then deontic logic has got some serious problems.

But I am saying that the inhabitants of world w can conceive of a morally perfect world where ~P. That's just what the accessibility relation is; if some world w can access some world x, that just means that the happenings of world x are conceivable from the perspective of world w. The same goes with the accessibility relation in deontic logic, with minor variations: if some world w can access some world x, that just means that the happenings of world x are morally conceivable from the perspective of world w.

I don't just say '~P' full stop, that would be like saying "◊~P at world w, therefore ~P at world w". What I'm saying, rather, is "◊~P at world w, therefore there is some other, possible world x where ~P is true, and which is morally conceivable from from the perspective of world w".

Look, suppose it is impermissible to murder someone from our point of view in the actual world. Does this mean that no one does in fact murder anyone in our actual world? Of course it doesn't, people murder other people all of the time in the actual world, in spite of that fact that murder is not morally permissible. However, suppose that we conceive, in our minds, of an imaginary, morally perfect world, where no one does anything that we actual-worlders consider to be morally impermissible. In this imaginary world, would anyone commit murder? No, because we have just said that murder is impermissible, and no one does anything impermissible in this imaginary world of ours.

◊~P at some world w means that ~P is permissible from the perspective of people living in world w; that means that the people of world w can conceive of an imaginary, morally perfect world called x, where ~P does in fact happen.

[Wyman]

ut I am saying that the inhabitants of world w can conceive of a morally perfect world where ~P. That's just what the accessibility relation is; if some world w can access some world x, that just means that the happenings of world x are conceivable from the perspective of world w. The same goes with the accessibility relation in deontic logic, with minor variations: if some world w can access some world x, that just means that the happenings of world x are morally conceivable from the perspective of world w.

I don't just say '~P' full stop, that would be like saying "◊~P at world w, therefore ~P at world w". What I'm saying, rather, is "◊~P at world w, therefore there is some other, possible world x where ~P is true, and which is morally conceivable from from the perspective of world w".

See, you snuck it in again - nothing about a proposition's being conceivable implies that the proposition is true.

In other words, we may hold an argument to be false (at least in world x) even when it is sound and valid. But we have supposed that 'a ⊢ ~□P' is such an argument. We can therefore valuate its corresponding conditional, 'a →~□P' as being false at world x within our model. For the conditional 'a →~□P' to be false, 'a' must be true and '~□P' must be false. The falsity of '~□P' is definitionally equivalent to the truth of '□P'. It follows that at world x:

You make a similar leap here. You say that because notP is conceivable in x, it follows that (not only that notP, see above) there is an argument in x that is sound but false. You correctly say that 'we may hold an argument to be false...' but then disregard the 'may' and leap to the conclusion that because one person in x may be illogical, therefore 'we' will hold a sound argument to be false and conclude not only that the squareP is possible, but that it is in fact is true.