Sep 8, 2011

Free Will and Determinism, part 7

Part 2:
Part 3:
Part 4:
Part 5:
Part 6:

It should seem that modeling perfect foreknowledge should be easy. After all, whatever is known is true, so setting up an equality between what is known and what is ture should be it.

Unfortunately, this is not enough. Truth can change. Consider any event. Prior to the event I can know that the event will happen. During the event I can know that it is happening. After the event I can know that it did happen. While these three statements of knowledge all refer to the same event, they are logically distinct states. Since we are explicitly dealing with temporal considerations, we need to factor in this transition. For the purposes of this discussion, we will consider two points of view with regards to a true proposition: knowledge of the proposition in the state in which it is true (current knowledge), and knowledge of the proposition in the state prior to the state in which it is true (foreknowledge):

fK(x) - foreknowledge of x; I know x will be true in the future in the next state.
nK(x) - current knowledge of x; I know that x is true in this state.

To formalize the relationship between these two types of knowledge, we will say:

If, in the current state, si, there is foreknowledge of some proposition x, then x is true in some state sj and (si,sj) is in R (a valid transition). The reverse is also true. If x is true in the current state, sj, and (si,sj) is in R (a valid transition), then I have foreknowledge of x in si.

Now we tie this in with our concept of choices discussed in the previous section:

If I am presented with a choice between A & B in the current state, si, then, also in this current state, I have either: foreknowledge of A and not B; or foreknowledge of B and not A.

Combined with our previous definition of Free Will, we have the following three definitions:

From this we construct our argument.
1. Given: I have a choice between A & B in the current state, si.

2. Applying defintion #2 to our given, I have either: foreknowledge of A and not B; or foreknowledge of B and not A.

Case 1:

a. I have foreknowledge of A and not B (in the current state, si).

b. I do not have foreknowledge of B

c. Since, according to definition #3, foreknowledge of a proposition entails the truth of that proposition in some state where that state can be transitioned from the current state, and vice versa. Thus, since I do not have foreknowledge of B, either B is not true in any state, or the state(s) in which it is true cannot be transitioned to from the current state.

d. Since we cannot transition from the current state to a state in which B is true (either because B is not true in any state, or it is true in a state that we cannot transition to), then our requirements for definition #1 are not met, therefore either we have not been presented a choice with possible outcome B, or we do not have Free Will regarding that choice.

Case 2:

Case to follows the same pattern as Case 1, swapping B with A, with the same conclusion:

Since we cannot transition from the current state to a state in which A is true (either because A is not true in any state, or it is true in a state that we cannot transition to), then our requirements for definition #1 are not met, therefore either we have not been presented a choice with possible outcome A, or we do not have Free Will regarding that choice.

Continuation from Main argument:

3. Since all possible cases result in the same conclusion, that conclusion is true, therefore we either have not been presented with a choice, or we do not have Free Will regarding that choice.

4. We have been presented with a choice (Given)

5. Therefore, we do not have Free Will regarding that choice.

[Click Pictures for Larger Image]

To recap:
If we define Free Will of a choice in terms of whether or not the outcomes of that choice are logically accessible, and divine knowledge as having foreknowledge of only one outcome, then we see that said foreknowledge restricts which possible paths are accessible. Forknowledge of one outcome makes all paths leading to the other outcome inaccessible and, thus, eliminates Free Will.

Again, this is an incompatabilist argument, it only shows the incompatability between Free Will and a form of determinism. In this case, divine foreknowledge. It makes no claim as to whether or not Free Will or divine foreknowledge exists, just that they cannot coexist.

That is not to say that this argument is without significance. Any world view that insists on Free Will and divine foreknowledge existing together is not logically coherent. It rests upon some premise that is not correct. This applies to many interpretations of Judeochristian religions. Our status as sinners deserving of punishment and unworthy of salvation stems directly from the fact that our actions perform of our own Free Will. Furthermore, part of God's divine nature is his omniscience, the fact that he knows all. These intepretations are invalid and must be abandoned.


Democurus said...

Ugh. There is an error in Definition #1. All the logical connectives on the right side of the arrow should be conjunctives (ANDs, /\). Fixing this would require a rewrite of this post due to the way images are uploaded, but I will do it at some point in the future. The rest of the argument holds, and treats the Definition properly (most notable step #6).

Democurus said...

Also, Step 5 should say "Disjunction Elimination"

Yolita said...

Hello, Democurus.
I read all your posts about Free Will and Determinism a few months ago and I found them very interesting. I must say I loved your discussion with B. von Traven and had a few chuckles reading it. I particularly loved his very debatable assertion that "Philosophy is not hostile to common sense". :-)
Needless to say, I totally agree with you that in order to study the relationship between two propositions one is not required to address the truth of either. And in fact, I think the kind of analysis you did is very fruitful to argue against certain philosophical systems, by showing that their assertions are not compatible.
In fact, Godel's famous Incompleteness Theorem can be (loosely) stated as: "No axiomatization of Peano Arithmetic can be both complete and consistent". With that he destroyed Hilbert's program.
I have a few comments to make, but first I must find out if I can actually post here, because I don't want to write a long comment only to lose it.

Yolita said...

OK. So that was fine. :-) So here it goes.

I have the feeling that we didn't get here all the posts that you intended us to get. Part 3 was completely empty and in Part 5 you mentioned that modal logic was previously discussed when it wasn't .It didn't really matter. I was very impressed with the clarity with which you presented the concepts involved, the arguments, the counter- arguments. All that was very nicely presented, very orderly, very easy to understand.

I want to make a few comments on the formalisations you do at the end. You say that you tried to use modal logic but that you found it to be insufficient. I got the impression that you have some misconceptions about modal logic. You say, for instance:

"Modal logic only talks about a finite number of propositions and whether they are: true in the actual world, true in some possible world, true in all possible worlds."

That really is not true. One normally does modal logic with an infinite set of propositional variables, it could even be non-countable. Your phrasing "modal logic only talks" is a little loose. In any formal theory of logic one has two levels: the theory and the meta-theory, the language and the meta-language. This will be important a bit later on.

You also mention that you will add another component by relating states to each other via a binary relation R. This also happens in modal logic. You seem to think that modal logis is fixed and can only talk about necessity and possibility. Not so.

I don't see, in anything you did, that you needed any particular logical theory, except the standard classical logic that we use when we prove theorems in ordinary mathematics. You used a few mathematical objects to model your arguments, but I didn't see any particular logic being needed.

It is important, though, to make the difference between a formal language and a metalanguage, because I feel you mixed them at the very end.

So, if I may, I'll give you a brief account of formal logic. How it works form a more philosophical point of view (I have the feeling that you must be a mathematician or a computer scientist).

I'll do that in another comment. This window is too small.

Yolita said...

In the late 19th century certain paradoxes in mathematics appeared which made it necessary to formalise the concept of mathematical proof. The language was completely formalised, a definition was given of what constituted a proof and a program to formalise the main mathematical theories was launched. That was the origin of all the contemporary theories of formal logic that we currently have.

Any study of any formal logical theory has two aspects: the syntactical and the semantical one. The syntactical aspect deals with a formal language, certain formulas are chosen as axioms, certain inference rules are given and one proceeds to prove theorems based on those axioms. The main concept there is the concept of provability. A theorem is a provable formula. The semantical aspect involves the concept of truth. One defines the possible interpretations for that formal language, one defines what it means for a formula to be true in a given interpretation. A tautology (or logically valid formula) is a formula true in all possible interpretations.

It turns out that we have logical systems for classical logic such that their theorems are precisely the logical truths. These are very good systems.

When one does modal logic, one is concerned with formalising the concept of necessity and possibility. One has a formal language which is the same as the ordinary first order logic with a symbol that represents necessity and a symbol that represents possibility. One studies this syntactically and semantically.

Syntactically, one looks for reasonable axioms and proves theorems. There are different theories with different axioms.

Semantically, one defines interpretations. And these interpretations are consitituted by a set of "possible worlds" (very similar to your states) each realising different propositional variables with a binary relation (called the accessibility relation) that relates some worlds to some others. One can have any binary relation R, but what is interesting is that different modal theories can be expressed by demanding certain properties of the binary relation R. For instance, R could be reflexive, symmetric, transitive...

What's important here is to make a difference. If one is proving a theorem in a certain theory T, one needs to produce a proof of the formula using the axioms of T and the rules of inference of T. And the theorem has to be a formula of the language of T.

In the metatheory, one can talk about formulas and their truth in different interpretations, and one can make statements about them and prove them as one would normally prove statements in mathematics.

Those are two different things, not to be confused.

I think there was a confusion of these two aspects in your work. In my next (and last) comment I will explain myself.

Yolita said...

When you said that modal logic was insufficient, I assumed that you would require more axioms than the ones normally found in the different modal theories. But that's not what you did. You only needed classical propositional logic in your last argument.

What you did was use certain mathematical objects to help you model certain propositions in ordinary language.

You introduced some symbols (like C(A,B), or fK(x), without being really formal about the language. Then you mixed the states si, sj, sk... and the binary relation R and defined what it means that C(A,B) is true or not. So you mixed the syntax with the semantics.

If you were doing a formal logic, you would need to explicitely list all the symbols of the language, give a definition of a well formed formula and then define the possible interpretations and define truth.

This would be too complicated and totally unnecessary for your purposes.

My suggestion is that you be a little more careful with the formalisation and also with the structure of the states. For instance, you didn't specify, if (, (sj,sk) both belong to R, does this imply that (si,sk) also belongs to R? Do you have infinite chains of connected states?

I can't read the formulas you wrote, but it seems to me that the definition at the end of Part 6 would need existential quantifiers for the states and a conjunction, as you pointed out. Also, definitions are usually written with a double-conditional (if and only if).

It doesn't look as if you needed current knowledge, nK(x). But I couldn't read the last proof, the images are very blurred. I think it would be better to talk about divine foreknowldege instead of saying "I have foreknowldege of A". It would make it more intuitive. Because as it is written, it is not clear why, if I am given a choice between A and B, I must have foreknowledge of A and not of B, or of B and not of A.

I think the argument presented at the end, with words, of why there is no free will in that choice if there is divine foreknowledge is very clear. I don't see the need to make a "formal proof" which is not really a formal proof in the contemporary sense of the word.

Having said that, I'll be happy to read it if you send it to me, because I really couldn't read it.

I really liked the way you presented this argument. Unfortunately, as it happens with many arguments, I don't think it would convince anybody who believes in free will and divine foreknowldege. But what do I know? I don't usually debate those issues.