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Saturday, 14 July 2007

Formal logic, counterfactual conditionals, and free will.

Formal logic seeks to abstract general principles of reasoning which are in some way independent of context. For example, if we know that sentence A is true, then we know that the sentence "Either A or B is true" is also true. We don't need to know exactly what A (or B) actually say in order to know that this is a valid piece of reasoning, where "valid" means that if the initial assumption (premise) is true, then the conclusion follows logically. We can write this as follows:

Premise: A is true.
Conclusion: Either A or B is true.

Here are some more valid arguments:

Premise: A and B are both true.
Conclusion: A is true.

Premise: If A is true then B is true.
Premise: A is true.
Conclusion: B is true.

Premise: If A is true then B is true.
Premise: B is not true.
Conclusion: A is not true.

Some people find this sort of thing more obvious than others. Well, all right, everybody finds those first few examples that I have given obvious, but the last one requires different amounts of squinting with your head to one side, depending on natural ability and how much practice you've had. If you can't see it immediately, suppose that both premises are true and note that if A were also true, B would be true (by the previous argument), which would make B both true and not true. As a result, A must be false.

If this all seems like so much obviousness, consider that around the beginning of the twentieth century there was a movement to construct all of mathematics from this sort of abstract logic. That this movement ultimately failed in an interesting way does not change my point that formal logic can go well beyond the obvious.

People use different types of shorthand for various logically important words like "and", "or", "not" and "if". I shall use the following:

A_______A is true.
~A______A is not true.
AvB_____Either A or B is true (or both)
A&B_____Both A and B are true.
A->B____If A is true then B is true.

One way of explaining the exact content of each of these statements is in a truth table such as the following:


The first line of this table states that, in the case where statements A and B are both true, the statement "~A" is false, and the statements "AvB", "A&B" and "A->B" are all true. Alternatively, we can look down the second column for the right to see that the statement "A&B" is only true in the case where both A and B are true.

Some of you may be slightly discomforted by the "A->B" column. I know I was, when I first saw it. Yes, the fact that today is not Wednesday means that the statement "If today is Wednesday, then I have five legs" is true. Certainly the truth table thinks so, anyway. Suppose A is the statement "Today is Wednesday", and B is the statement "Lynet has five legs". We are given that A is false. According to the bottom two lines of the truth table this means that "A->B" is true. On the other hand, suppose today is Wednesday, and I do not have five legs. Now the truth table tells us that "A->B" is false.

Now, I know you can't see me over the internet, but I can assure you that I never have five legs. So the statement "If today is Wednesday, then Lynet has five legs" is true unless it is Wednesday, in which case it is false. Got that?

In mathematics, this version of "if . . . then" works fine, because things which are mathematically false stay false; things which are mathematically true stay true. However, in real life we often want to know what things would be like if they were different. If today was Wednesday, would I have five legs? This turns out to be a very different sort of question! An "if . . . then" statement is called a conditional statement. An "if . . . then" statement in which you are speaking of what would happen if something were true is called a counterfactual conditional statement.

Counterfactual conditional statements are pretty well impossible to put into formal logic. They seem to have an inextricable contextual component which makes them impossible to describe in a contextless, abstract way. Whereas the logical "A->B" doesn't require us to know anything about what A or B actually say, the statement "If A was true, B would be true" really needs context, and some idea of what A and B are, before we can use it. This is because we are speaking of some other possible (or sometimes impossible) world where A is true, and we need context and the elusive 'common sense' to tell us how much of the current world to imagine as changed before we consider whether B would be true in such a world. Consider the following exchange:

"If I had my cousin's money, I'd be a happy woman."

"If you had your cousin's money, you'd be in jail."

It's possible for both these statements to be true in some sense, even if the first speaker would most assuredly not be happy in jail! The first statement is considering a possible world in which the speaker both has her cousin's money, and has a legal right to her cousin's money. The second statement changes the world just enough that the first speaker would have her cousin's money, but does not change the fact that the first speaker has no legal right to her cousin's money. How much of the world do we change in evaluating this sort of statement? It depends on context. Sometimes it is obvious and sometimes it isn't. Either way, we can't substitute abstract As and Bs for the statements given. We need to know what is actually being said if we want to have any hope of deciding what is meant.

Counterfactual conditionals are, I would say, absolutely fundamental to human thought. However, the fact that they don't have an exact logical meaning means that they can interact with more logical ways of describing the world in an interesting fashion. For example, consider the question of free will. Most people would, I suspect, agree that we have free will in a situation if we could have acted differently. That's a counterfactual conditional right there, and the question of what free will means can twist in exactly the way that counterfactual conditionals do. For example, sometimes a person might say "I didn't have a choice about handing him my wallet -- the man had a knife to my throat!" Now technically, you could choose not to hand over your wallet. You could die instead if you wanted to. Of course, you'd lose the wallet anyway, so it would be a rather peculiar choice, which is why most people would accept that statement without a blink. The world where the wallet is taken from your dead body rather than given by you isn't considered as a relevant possibility when evaluating whether you could have acted differently.

Do we have some sort of 'free will' beyond the fact that we could act differently if we wanted to, counterfactually speaking? I find it hard to imagine how such a thing could work. That doesn't mean such a thing doesn't exist, of course, but I'll withhold belief until I'm given evidence and/or an accurate explanation of what such free will would actually mean.


Halfmom, AKA, Susan said...

OK - I know that I hate math and you don't - and that I am taking a brain burnout break so my brain just isn't working terribly well - so take this as tongue in cheek with me shaking my head incredulously

but why in the world would you choose this as a post? it sounds much more like a thesis proposal

if you are looking for a theological answer - there are many - but I will have to understand more about exactly what you are asking - and sleep more - to be able to formulate any kind of logical answer

Lynet said...

Is it really that complicated? I'm trying not to blog flat out about atheism (I know I thought of this because it has theological implications, but its impact on philosophy goes well beyond that).

I've blogged about maths in the past and will probably do so again; feel free to ignore it in most cases. This case is a special one, of course, because it's a case where logic interlinks with observations about the way we, as human beings, perceive the world.

I'm not demanding any answers, just explaining something that affects my view of the world. You're free to comment, critcise or ignore as you wish :-)

Halfmom, AKA, Susan said...

Yes, theologically, at least for me, free-will is very complicated and I am simply more aware at this point in time because I've been doing quite a bit of ruminating on it lately.

Although I certainly do not know you as a person, you do not seem to me to be of the demanding sort - more of a persistant questioner who is rather determined to settle things in her own mind - a pattern familiar to me. Do your parents perhaps tell you that most of you early sentences had "why" in them?

It is a very valid question - certainly theologically and I would imagine mathmatically, however I have already confessed to being more biologically than mathmatically gifted. Perhaps I shall have a chance to pass your blog entry on to a friend who actually is a mathematician - quite literally because he is chairman of the Math and Physics Depts at a University not far from here - and he can give you a more logical answer than I.

Best wishes for your day - although I would imagine it is partially over as I believe we are in quite different time zones. I still have many grants hours left ahead of me and am almost at the weeping stage of tiredness now - you should be prepared for this as it becomes and ebb and flow of graduate school days! I'm afraid.

Alon Levy said...

The logical if-then operator isn't really if-then; it's just a convenient way of combining negation and disjunction. In Loglan, the word for "implies" is even constructed regularly from no and or.

On another note, are you even into mathematical logic, or is it a philosophical idea that's unrelated to your mathematical interests?

Lynet said...

I used to be more 'into' mathematical logic than I am -- I read some Raymond Smullyan while I was in high school, and took a couple of courses when I got to uni. These days, it's only related to my interests insofar as you can find links between all aspects of methematics.

Persistent questioners can be demanding, you know, Halfmom. But yes, I definitely fall into that category.

Halfmom, AKA, Susan said...

I suppose I do not equate demanding and persistent - one having a negative connotation in my book and the other a positive one.

I am passing your blog site entry onto my math professor friend who hopefully will have time to comment as he can speak to both the math and the underlying philosophical/theological issues.

You'll like him - he's about as nerdy (did you see U-tube's "white and nerdy" - such is our beloved Tom!!) as they come - and kind, gentle and patient, even with ever persistent me!

L.L. Barkat said...

Okay, so can you give me a practical example, like the Wednesday and Lynet has five legs thing?

For instance, I used to have a very brilliant friend and we'd have conversations like this...

He'd say, "if I step off the curb, but God knew I was going to, does that mean I had free will to do so? Did the fact that God knew it mean I had to do it?"

Of course, I just smiled and said, "Gee, B, I don't know." And realized this could make even the simplest of walks a complete nightmare for poor B, as he tried to decide if he was a puppet or a free agent there on the street.

So, yes, examples are good. And so are putting our fears out on the table. Here was mine, and B's non-stop questions actually expressed it...

I always associated God with "male"...males had been the abusers and neglectors in my early life... therefore (see my great logic language here?)... therefore, God must be a secret abuser or neglector who wants to control and damage me.

Why, you better believe, I needed free will even if it didn't exist. (Which I actually think it does.)

Lynet said...

Gosh, LL. You would need free will, under those circumstances (at the very least!).

An example, huh? I'll try. Suppose you were deciding whether to have peanut butter on your toast. In so doing, you have a perception of yourself as an autonomous being, able to go one way or the other. However, you are also, yourself, a physical object, so to speak, subject to physical laws and behaving accordingly. On a slightly less fundamental level, you're a person subject to habits, desires and so forth, and whatever decision you make will be largely based on that. The question is, is your decision simply a matter of physics and/or preferences taking their course, or is there some part of you that really does act 'freely'? And if you act freely, does that mean you do so for no reason? Because if it's done for a reason, then presumably you are controlled by whatever it is about you that cares for that reason; you could act differently if you cared differently, perhaps, but that's a completely different situation, with you being controlled by a different set of cares.

I say "controlled by" in the very loosest sense; I am, of course, speaking of being controlled by yourself. But surely there's a level on which you can't control who you are?

L.L. Barkat said...

Surely. For starters, I'm a woman, a mother, a spouse, a writer. I have some control over these things and they have some control over me. I wonder sometimes if the "control" question is the wrong question. I wonder why I and others get so hung up on it.

Halfmom, AKA, Susan said...

I think that LL's question is one I'd love to hear you blog about Lynet - why do we get so hung up on control? Sometimes I feel like it's not as if I really had any, anyway! And yet, control seems very important and to determine how much control you have seems to me to be to determine how much free will you have to choose too.

Lynet said...

Well, trouble is, I don't think we have any 'ultimate', uncaused, control -- and I don't think that's a big deal, either!

Frankly, I have no idea why people get hung up on control.

Theologically, of course, whether we have any ultimate control -- control of ourselves that is outside of the things God set up -- is the difference between, for example, Calvinism and more standard protestantism.

In other words, it's the difference between "God sets us up with initial conditions such that we will do bad things and then punishes us horribly for it" and "God sets us up with initial conditions such that it is theoretically (albeit apparently not practically) possible for us to behave ourselves, depending on some ultimate choices made by us, and then punishes us horribly for not always making the right choices."

Some people think the second one is significantly nicer.

I'm not going to put this in a blog post because Hell is a sick, sick doctrine and I'd rather write about nicer things. That post on 'doubt' has probably already exceeded my quota of righteous indignation for the month.

Ted M. Gossard said...

Interesting post, and while I love philosophy up to a point, and theology, I don't even minor on math. (more of a philosophy wannabe; I see theology as everything having relation to God and therefore everything being important or significant in some way)

I believe that humans do have free will which though certainly in parameters is a part of the image of God in them. Therefore I'm of the persuasion that one does not HAVE to act in a certain way, and am doubtful that there is a coercion to simply act in the one way due to intrinsic and extrinsic factors.

And on "time", I'm not sure God knows what I'll do an hour from now. I'm not convinced that time future is present to God. I do believe that God will make all happen in the future as God has decided and as God wills; in that sense God does know.

Larry Hamelin said...

To get the sorts of counterfactual mappings to our intuitive notions of if... then..., you have to use modal logic.

Lynet said...

Even modal logic won't take you all the way there, surely -- counterfactuals include references to impossible worlds sometimes and, as I've pointed out, the exact possible world(s) referred to are highly contextually dependent.

Mind you, I can see how you could get a version of 'if..then' in modal logic that was a lot closer to the intuitive version. You still have the problem of 'how much about the world is allowed to change to cause the counterfactual proposition to come about', though.

Alon Levy said...

The problem with intuitive causal arguments in formal logic is that formally, there is no such thing as causation. Even in physics there's no such thing as causation; the universe is simply a static object in spacetime, and we're just worldlines in it. I know A Brief History of Time isn't a good guide to anything, but there Hawking has to define the distinction of past versus future by thermodynamics and cosmology rather than causation.

That, I suppose, why formally you have to fall back on defining if A then B as B or not A...

Lynet said...

Well, cause is another thing again, isn't it? I wrote a philosophy of science paper on causation a while back. Even if you just allow counterfactuals as-is, causation can be a tricky thing to define.

This probably casts doubt on 'every effect must have a cause' as an axiom from which to reason.

Anonymous said...


What do you think of Frankfurt-cases?

Suppose Murderer wants to kill Victim. Suppose that there is a Manipulator who can monitor Murderer's brain activities and manipulate Murderer with regards to these neural activities. If Murderer shows the slightest doubt about killing Victim, Manipulator will step in and excite Murderer's brain in such a way to ensure that Murderer murders Victim.

Murderer goes ahead and kills Victim. Manipulator never had to intervene.

Now Murderer couldn't have done otherwise, but he still killed Victim freely and is responsible for his death.

Lynet said...

Interesting example. It obviously doesn't affect anything beyond definitions, I mean, it's not like anyone has any doubt about whether Murderer is responsible.

Perhaps Murderer could have done otherwise in some sense. Having someone aler your brain so that you kill someone is 'otherwise' to killing someone with no brain alteration. Am I cheating, there?

Anonymous said...

Great post, hope people are still reading this. Anyhow, I thought of the following scenario:
I program an agent (computer program/robot) who's sole "purpose" in "life" is to wonder around in an ideal world. Usually all it does is just keep on going in a straight path defined by clear boundaries. Suddenly the path splits. The path on the right is clear and the path on the left is blocked by some unmovable obstacle (I forgot to mention the agent has a sensor allowing him to sense his environment, and so he sees the obstacle). The agent chooses the path on the right and keeps wondering "happily" on and on...
What is the meaning of such a choice? Is it any different from decisions humans make? Does the robot have free will? any will?
My view is that that we are no different in terms of the 'free willingness'. You might say "heck no!, you, the programmer could know to the very fine details what this agent will do", meaning to say the robot had little to do with his decision. I say, well so is the case with us humans. All our actions are ultimately determined by the laws of physics and the particles which obey these laws.
But the analogy doesn't end there. I agree with Lynet and think the truly important view is that which ponders "hmm... well what if the obstacle were on the right path!" (notice the counterfactual?). Well, if I'd be a better programmer I'd say that my clever agent would immediately come on to our little scheme and choose the
path on the left!. (being aware of my programming skills, my agent's behavior might be a bit more unpredictable...).
And so it is with us humans, we are able to put ourselves in imaginary situations, guessing both what are the relevant situations to be imagined and what is the best course of action. There lies the true mystery!