Formal Proof About Classical Free Will - Page 5

hp.Shell

United States2527 Posts

May 29 2014 05:22 GMT

#81

On the subject of breathing, I know that it is impossible to kill yourself by simply holding your breath. You will first lose consciousness, at which time your autonomic nerve system retakes control of your breathing. Thus, the instant you lose consciousness, you will have an inhalation reflex.

Cool stuff. I think we have free will, logical determinism be damned.

spinesheath

Germany8679 Posts

May 29 2014 05:46 GMT

#82

On May 29 2014 09:09 Lixler wrote:
TL;DR of the following: for your reductio, you assume that the antecedent of the conditional is true. but the conditional can be true without the antecedent being true. if we assume, as we logically ought to, that our conditional as a whole is true, no contradiction follows. you have simply proven that the antecedent must be false (viz. that B could not be impossible)

let's fix your formalization somewhat: instead of using B, we'll just use ~A. we'll also introduce quantificational and modal symbols

so classical free will is:
a -> ◇~a
or more properly, since classical free will only says that some of our actions are freely willed:
Ǝa (a -> ◇~a)

the contrapositive (for our assumed a) is now
~◇~a -> ~a
or
□a -> ~a

we can conclude a few things from this, but first let's run through your logical proof with our new symbols.

our assumption, in attempt of the reductio, is that ~a was not possible, or rather that □a.

from □a -> ~a we conclude both (this is trivially true)
a
~a
and obviously this is a contradiction.

well, what went wrong? this: in attempt to prove that your conditional led to a contradiction, you assumed that the antecedent was true. but we don't assume the truth of our antecedent in order to prove that our conditional was false. we assume that our conditional was true! and there are three ways for our conditional to be true. they are, as follows
□a ^ ~a (which leads to a contradiction)
~□a ^ ~a
~□a ^ a

now our second set of truth values would lead to a contradiction, since we already assumed that a (we assumed there was some action such that we could have not done it). we are forced to conclude that ~□a ^ a. unfortunately for your proof, these two values do not lead to a contradiction.

Look, I really don't know this notation and didn't bother trying to figure it out, but I think it's just saying this:

On May 29 2014 06:17 spinesheath wrote:
So you have a contradiction if you assume that there is some B that was not possible.

Then obviously everything is possible.

231 Posts

May 29 2014 11:47 GMT

#83

Lixler pointed out the major problem with your proof - namely, you're using first order logic, not modal logic. But if you do try to rewrite it in modal logic, I'd be wary about using contraposition. Ernest Sosa has a famous argument against contraposing subjunctive conditionals, which I think applies to your case as well.

bookwyrm

United States722 Posts

May 29 2014 17:38 GMT

#84

^Yeah! It's easy to show that possibility can't exist when you formalize your proof in a language that has no marker for possibility!

Boblion

France8043 Posts

May 29 2014 20:36 GMT

#85

Hello Ludwig.

Deleted User 137586

7859 Posts

May 29 2014 21:01 GMT

#86

On May 29 2014 09:29 MichaelDonovan wrote:

Show nested quote +

On May 29 2014 08:39 Ghanburighan wrote:
Interesting stuff, but here are a few issues:

a) You're using classical propositional logic to discuss possibilities, you'd need modal logic for a better representation. "Recall that we have assumed that B is not possible. Since B is not possible, we deduce: (4) ~B*." You can't do this formally without modal logic. And if you use modal logic, make sure you pick a system in which ~Diamond B* entails ~B*.

b) If you do adopt a modal logic, the original thesis would be better described as "at some point (world, time, etc.) where Diamond A holds, Diamond B holds as well."

c) I'm assuming you're utilizing material implication for "->" which has many known puzzles and problems. I don't think we can say much about free will or other concepts in a system in which B makes true A->B, i.e., where sitting in your office, makes true the statement: if you were blown up by a bomb this morning, then you're sitting in your office.

d) To attain (5) you make use of contraposition, but this rule doesn't correspond to our view of rational reasoning with probabilities. Consider Grice's Yog and Zog puzzle:

Yog and Zog are playing chess with special rules. Yog gets white 9/10 times and there are no draws. They have already played around 100 games, and Yog emerged victorious in 80 out of 90 of the games in which Yog had white, but Zog won all the remaining games. Now, the following two sentences have different probabilities.

a. If Yog had white, Yog won.
b. If Yog lost, Yog had black.

The probability that the sentence (a) holds is 8/9 but it is only 1/2 for sentence (b). The problem with this situation is that (a) and (b) are equivalent if analyzed as material implication. This is because when you play chess, you use either the white or black pieces. So, playing with not white pieces is the same as playing with black pieces. And losing is the same as not winning when draws are taken out of the rules of chess. So if (a) is represented by p -> q then its contraposition ~q -> ~p is (b). But equivalent sentences should not have different probabilities, 8/9 and 1/2, respectively.

a) It is at this point in the proof where I am able to get rid of the need for any modal operators. I only need ~<>B* to show ~B*. And yes, I should have included that somewhere in my proof, but I am assuming that B being impossible entails that B* cannot be true.

b) Sure, I don't see any problem with saying it like that. Modal logic gets kind of messy though sometimes, so I wanted to try to do away with it.

c) This objection would be problematic if it were not for the fact that all of my statements in this arguments are based only on the logical structure of the proposition. That is, we don't really care if it makes sense empirically. What we're showing is that the classical free will thesis is logically incoherent. This doesn't say much about free will. It just says that the way the classical thesis is written leads to contradictions.

So, the fact that being blown up this morning makes it impossible for me to be sitting in my office isn't really important. If it were, we would include that statement in our argument, and the contradiction would be clear. I'm not sure if this a sufficient response to your objection though, so hammer it a bit more if you're dissatisfied.

d) That's a fun example. I'll have to think about that for a bit. But I don't think it's a problem for my argument because we either have free will or we don't. We either do action A or we do action not A. It's not like If you do action B, then you probably didn't do A. Or something like that. When you have mutually exclusive sets, the probability of their intersection is zero since their intersection is the empty set. I dunno if that addresses the problem completely, so again, hammer it some more if you're not satisfied.

Let's address (b) as that's the most important one. Diamond A wedge Diamond B isn't a contradiction, you won't be able to derive Diamond B and ~Diamond B from it unless you introduce a formula which is a contradiction itself. But then the problem is with that assumption and not free will.

Basically, as others have said, your formalism fails because you're allowing moves which modal logic would not allow.

***

Regarding (c), I can say that free will is linked to causation. We can either influence the chain of events or we cannot. If you represent causation with material implication, you can derive things which we know do not correspond to causation itself. This is basically the same as (d), as the logical formalism makes counter-intuitive predictions about causation, we should identify the problem with the logic, and not with free will. And besides not using modal logic, the biggest problem is the use of material implication instead of more refined accounts of implication such as strict implication, relevant implication, or Kratzer's context-sensitive conditionals.

radscorpion9

Canada2252 Posts

May 29 2014 22:15 GMT

#87

This was a really cool blog, I like practising my logic skills so it was fun to follow along the discussion. To be honest I didn't follow Lixler's analysis that well either because I'm not used to the symbols and am not clear on what they mean, but I think I also noticed an error in your proof and would like to point it out in more readily comprehensible terms.

I think the biconditional statement needs to be modified, because as it stands it assumes that A* and B* are both possible. But the contrapositive of your proposition indicates that if B* is impossible, then A* is impossible. So there should be the possibility that A* and B* are both impossible.

So to be clear, B is any distinct action that is not A. The biconditional should be modified to this:

If B and A both exist as possible actions such that B* and A* are both possible, then:

(3) B* <-> ~A*

This way it makes sense, because B needs to be possible before an agent performs the action (B*), and the contrapositive requires that A be possible before A* is.

So then when you start with the assumption that B is impossible, you do indeed get ~B* as a consequence, but before you continue on to using the biconditional I think you would need to establish that A is actually possible (and consequently A*), which you can't really get from anything else unfortunately. Thus it would be invalid to use the biconditional in your fifth step, and the argument just ends there uncompleted.

So that's the main thing: The biconditional assumes both actions are always possible, while the contrapositive to your proposition considers the possibility that both may be impossible. The biconditional thus needs to be modified to allow for these possibilities

edits: Grammar. By the way I hope you enjoyed your counterstrike session!! I used to love that game

Lixler

United States265 Posts

May 30 2014 04:18 GMT

#88

On May 29 2014 14:46 spinesheath wrote:

Show nested quote +

Look, I really don't know this notation and didn't bother trying to figure it out, but I think it's just saying this:

Show nested quote +

On May 29 2014 20:47 zf wrote:
Lixler pointed out the major problem with your proof - namely, you're using first order logic, not modal logic. But if you do try to rewrite it in modal logic, I'd be wary about using contraposition. Ernest Sosa has a famous argument against contraposing subjunctive conditionals, which I think applies to your case as well.

I don't think I'm saying either of these things. Zf's point is fairly obvious - we ought only deal with possibility using modal logic - but the exact problem I'm pointing out in OP's proof is not rooted specifically in his choice of logic. Specifically, he sets up his reductio falsely. Quoting it now for ease of reading:

Contrapositive of the proposition: If it was not possible for the agent to have done B, then the agent did not do A.

We will attempt to disprove the proposition by showing that its logically equivalent contrapositive leads to a contradiction.

So we assume (for reductio) that it was not possible for the agent to have done B. If the proposition is true, then it should follow that the agent did not do A. We will show that this is not the case.

When we attempt a reductio of some proposition, we assume that proposition is true and work to derive a contradiction. OP thinks he is doing this when he assumes "that it was not possible for the agent to have done B," but in fact he needs to assume "If it was not possible for the agent to have done B, then the agent did not do A." There are three possible ways for this latter conditional to be true. The first, which OP demonstrates is contradictory, is for it to have been possible for the agent to have done B and for the agent to not have done A. The second, which also will not work (as I demonstrated in the symbolization) is for it to have been possible for the agent to have done B and for the agent not to have done A. The third, which leads to no contradiction, is for it to have been possible for the agent to have done B and for the agent to have done A. When these two statements hold (i.e. when the antecedent of the conditional is false while its consequent is true), the conditional is true (as we assumed), yet no contradiction follows. So OP has failed to demonstrate that the classical free will thesis is contradictory.

The specific move where OP goes wrong involves a fairly common mistake made by people learning formal logic. Namely, he assumes that the only way for a conditional to be true is for both the antecedent and the consequent to be true. But this is not so. As it turns out, we can derive a contradiction from any conditional and its contrapositive if we don't assume this (specifically because the equivalence of a conditional with its contrapositive relies on the fact that the conditional is true when both its parts are false, but whatever). Let's take the following example.

If I love my wife, I will buy her a gift for our anniversary.
Contrapositive: If I will not buy my wife a gift for our anniversary, I do not love her.
Now let us assume 1) that a conditional can only be true when both its parts are true and 2) that our contrapositive is true. This means that we have to assume that it is true 1) that I will not buy my wife a gift for our anniversary and 2) that I do not love her. But if both these are the case, and it is further the case that a conditional can only be true when both parts are true, then it follows that our initial conditional is false. But this leads us to a contradiction, since the contrapositive (which we assumed to be true) is supposed to be logically equivalent to the initial conditional, i.e. true when it is true, false when it is false.

We cannot simultaneously maintain that a conditional is only true when both its parts are true and that a conditional is logically equivalent to its contrapositive. Since OP does this in his proof, it is invalid. Indeed, we could prove anything we wanted if we tried to maintain both these things. Hopefully it is clear now where and why OP's argument goes wrong, although obviously there are problems with his symbolization that don't have anything to do with my qualm.

MichaelDonovan

United States1453 Posts

May 30 2014 05:40 GMT

#89

Hey guys I just wanna say thanks for all the great comments. This blog turned out to be pretty fun.

EatThePath

United States3943 Posts

May 30 2014 16:45 GMT

#90

Ah well done explanation lixler, thanks. Also ^

Nemesis

Canada2568 Posts

May 31 2014 04:03 GMT

#91

On May 29 2014 10:22 2Pacalypse- wrote:

Show nested quote +

On May 29 2014 10:04 MichaelDonovan wrote:

On May 29 2014 09:59 2Pacalypse- wrote:

On May 29 2014 09:53 MichaelDonovan wrote:

On May 29 2014 09:19 2Pacalypse- wrote:

On May 29 2014 07:54 MichaelDonovan wrote:

On May 29 2014 07:50 2Pacalypse- wrote:

On May 29 2014 07:35 MichaelDonovan wrote:

On May 29 2014 07:31 2Pacalypse- wrote:

On May 29 2014 07:14 MichaelDonovan wrote:
[quote]

I would be wary of believing anything that seems to be "painfully obvious." If something like this were painfully obvious, we wouldn't be arguing about it for hundreds of years.

That's because philosophers love to argue. Until the real scientist comes along and puts the matter to rest with empiric evidence instead of pure thought

.

Facetiousness aside, you should really watch the video. Sam makes a compelling argument on why the very concept of free will doesn't really make sense.

Empirical evidence is hard to rely on when our sense perception is so limited

There are really two versions of the world: The world as we perceive it to be and the world as it actually is outside of our experience of it. Truth can be found in the world as it actually is. What we perceive is only contingent on the accuracy and completeness of our sense perception. The only way to get at the way the world actually is, it would seem, through rational thought and deductive reasoning. A fairly lofty task though, to be sure.

Empirical evidence is hard to rely on? Is that why science doesn't make any progress?

The best method that we have to find out truth about the world, an objective truth, is scientific method. Rational thought and deductive reasoning go out the window once we're confronted with something that doesn't seem rational, eg. quantum mechanics.

Well the problem is that empirical evidence isn't foolproof. We can only gather evidence from what we can sense. How do we know we're not in the matrix world, for example? We don't really know that do we? No amount of empirical evidence can prove that we aren't dreaming or that we aren't just brains in vats being stimulated with the illusion of experience. This is because any empirical evidence we gather is contingent upon our trust for our ability to sense it.

Quantum mechanics is not irrational necessarily. It just defies some our previous assumptions about the world. It renders many statements false that were once though to be true. That doesn't mean it's logically incoherent. It just means we don't understand it based on our current assumptions.

We can't disprove that we're not living in the matrix world. We can't disprove an infinite number of things. That's why we don't pay attention to things unless there's some evidence for them (well, except religion). It wouldn't be very reasonable to start thinking about things that we have no shred of evidence for (like if there's a teapot orbiting around Jupiter). Sure, you can lock yourself up in a room and try to use deductive reasoning all you want, but until you actually go outside of the room and look at what the world is telling you, you won't make any progress and in fact, most of your deductions will be simply wrong.

On May 29 2014 08:38 MichaelDonovan wrote:

On May 29 2014 08:36 ninazerg wrote:

On May 29 2014 07:14 MichaelDonovan wrote:

On May 29 2014 07:09 2Pacalypse- wrote:
A good video on free will by Sam Harris that made it painfully obvious (to me) that free will is an illusion:

http://www.youtube.com/watch?v=hzj1helboAE

I would be wary of believing anything that seems to be "painfully obvious." If something like this were painfully obvious, we wouldn't be arguing about it for hundreds of years.

So the length of time by which we argue something determines how obvious it may or may not be?

Determines is a wrong word. Indicates is might be better. What I'm saying is that if the guy in this video actually came up with a solution so obvious that you couldn't argue with it, then I would have heard about it by now, and there would be nothing more to argue. But since people are still arguing, his solution can't be all that obvious.

I did write it became obvious to *me*. This was not a formal scientific theory that Sam proposed, but rather a compelling argument that "makes sense" once you think about it. There are people who challenges his argument though, like philosopher Dan Dennett, but from what I've read it doesn't look like a very good challenge.

Also, people are still arguing about creation vs evolution so you can't possibly take the length of the arguing as an indicator to one's side validity. Perhaps just watch the video or read about Sam's arguments on free will.

I'm not saying that empirical endeavors are pointless. I'm just saying that you can't really value empirical evidence too highly and trust it too much.

And the only way of getting past the limits of our sense perception is probably to use logical deductions. You can't just go by empirical evidence for answers to your questions because your evidence is so very limited. That's not to say that logic alone can answer all your questions. Logic is merely a formal way of thinking. You have to fill the propositions with empirical assumptions on your own. And then any truth you derive is contingent upon those assumptions, which are contingent upon your sense perception.

And that's what scientific method is: rational thinking applied to empiric evidence.

Right, but the difference is that if empirical evidence suggests something that logic denies, then we conclude that our sense perception misled us. But if logic implies something that empirical evidence seems to deny, then what do we again conclude that our sense perception is wrong.

What I'm saying is that sense perception and empirical evidence are always the first suspects for error. Logic is more trustworthy.

Uhm, no, it's the other way around.

When we demonstrated with empirical evidence that the electron can be at the two places at once, something that seems logically impossible, we didn't dismiss it due to our sense perception misleading us.

Sure, you can try to "update" logic to fit with the reality, but you can't update reality to fit with your logic.

While I think that what he said is incorrect about dismissing empirical evidence over logic, quantum mechanics is not really illogical. It has its own rules which it follows, which just seems counter intuitive to everyday human experiences.

Logic is simply a way of connecting one statement to another. It's not that logic dictates that it's impossible for the electron to be at two places at once, it's the assumption of how a particle should act that is wrong. Initial assumptions from which the logic is derived can be wrong, but logic itself is not wrong.

But at the same time, logic is just a play on words and doesn't really apply to reality unless you actually have something like empirical evidence to start from which you know is correct in the real world.

What I'm talking about is really all kinds of logic such as mathematical logic, logical probabilism, and whatever other kinds of valid logic you can think off. If he's only talking about classical logic, then sure feel free to bash him

ninazerg

United States7291 Posts

May 31 2014 06:44 GMT

#92

On May 29 2014 09:45 MichaelDonovan wrote:

Show nested quote +

On May 29 2014 09:14 micronesia wrote:

On May 29 2014 09:07 MichaelDonovan wrote:

On May 29 2014 08:58 micronesia wrote:

On May 29 2014 08:33 MichaelDonovan wrote:

On May 29 2014 08:20 micronesia wrote:
It's unfortunate that I don't remember the mathematical rules of logic that well, but I don't see how you can use logic alone to prove the following:

It is not possible for every action to have at least one alternative, mutually exclusive action.

I don't think there are enough tools to work with. Putting technical lines of math aside, what is the reason why the above statement is true?

What you're getting at is the only non-formal assumption that I feel can be poked at in my proof.

My reasoning is as follows:

Recall that we are assuming for reductio that the classical free will thesis is true (we are then showing that a contradiction arises from this).

So since we are assuming classical free will exists, it would seem like no matter how pidgeon-holed into a line of action a person may seem to be, he always has the option of not doing that action, which is a contrary action in itself.

Here's why I say this:

Completing an action requires a break in inertia of sorts. That is, in order to execute an action, one must go from a state of inaction to a state of action and this requires some kind of effort, or at least a willing. If one does not will that the action be executed, they can choose not to act.

The way I think of it is like this: Imagine you are walking on a narrow path over an abyss. You cannot jump off the sides of the path because there is an invisible wall or something. And as you walk, the path behind you gets deleted such that you cannot go the other way. So it would seem as though you have only one choice, which is to continue walking forward. But it turns out that you still have the option of standing still.

I dunno. Maybe this is a weird example. But I think as long as we are assuming free will to exist (for reductio) we can assume that an action that you will should also have an alternative option if you were not to will it. It's easy to imagine a case were you are tide up with a sock in your mouth and you can't move, and say well, there seems to be no other option... But you have the option of willingly accepting your immobility or at least trying to escape (pointless as it may be). The action of accepting your captivity and the action of fighting it (regardless of the effectiveness of your struggle, which may be zero) are contrary actions.

Everything above seems to be an explanation supporting this classical free will thesis... following along using this plain-English approach, what is the contradiction?

The contradiction is probably something like this:

Taking a step forward is a sufficient condition for not having stood still.
Not having stood still is a necessary condition for having stepped forward.
Having stood still is a sufficient condition for not having stepped forward.
Not having stepped forward is a necessary condition for having stood still.

So if I chose to step forward, this makes it the case that I did not choose to stand still.
But not having chosen to stand still is the only way that it could have been possible to step forward. Thus, if you stepped forward, it must be the case that standing still was not possible.

Something like that.

Hm, when you put the 'contradiction' in plain English like this, it seems like your argument is teleological. As others have said, you are using your conclusion as evidence of your conclusion in a somewhat circular manner. Once again, I will not point to logical inaccuracies in your 'proof' as I do not have the necessarily knowledge to do so, but it seems a bit ridiculous to me to say:

"In order to choose an action, I had to not choose other actions, but it was necessary for me to not choose those other actions in order to choose the first action, so really I didn't have a choice," which is, in essence, what your contradiction is. It's almost a semantic argument rather than a logical one, when looked at in words rather than proofs.

Well... I'm not really satisfied with the way I worded it in plain English. It's kind of hard to say it properly because it's not all intuitive. Trying to explain why 1 + 1 = 2 in English without using formal logic to derive this truth from definitions leads to an explanation that won't really be satisfying and is probably full of holes. And I'm not sure I put the logic into words correctly there either...

Not having stepped forward is a necessary condition for having stood still. Was it possible for you to have stood still given that you stepped forward? No, because not having stepped forward is a necessary condition for having stood still. So given that you have stepped forward, it is impossible that you could have stood still.

I dunno.

All that tells me is that I can't change an action I made in the past, but has very little implications for what I'm going to do in the future, unless you mean to say that you're going to do what you do because you can't not do what you're going to do. But if that's the case, all that means is that I'm going to make decisions, and it is impossible for me to not make decisions, because if I make a decision not to decide, I'm still making a decision and cancelling out my decision to not decide anything.

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