Dmytro Taranovsky

October 9, 2002

Possibility and Impossibility

For an event or object, "it is possible" means that if it is then no contradiction would occur.

Example:  If x is a real number, it is impossible that x2=-1.  It is possible that x=1.  It is possible that x=2.

The world is what it is.  If event A will happen, then it is not possible for event A not to happen:  Assume that A will not happen; since A will happen, a contradiction is reached.

Some consider that there are many possible worlds, one of which is our world.  They define P to be possible if P is true for a possible world.  Event A is possible if there is a world for which it happens.  P is impossible if there is no world for which P is true.  Let P equal "event A happens in our  world".  If A happens for our world, then for every possible world, A happens in our world, so it is impossible for A not to happen in our world.  If A does not happen for this world, then there is no possible world for which A happens in our world, so it is impossible for A to happen in our world.

A Diabolical Offer

Suppose that someone offers you thirty dollars in exchange for the following:  The person will make a statement; if the statement is true, then you have to pay at least twenty dollars; if the statement is false, you have to pay exactly ten dollars (you cannot pay more than ten dollars).  Considering the offer to be attractive, you accept.  The person now states:  "You will either pay me exactly ten dollars or at least a million dollars".  If you pay ten dollars, then the statement is not false, which contradicts to the contract.  Thus, you cannot pay ten dollars, so the statement is required to be true.  But then, however, you will have pay at least a million dollars.  How do you resolve the problem?  (Assume that in your jurisdiction courts do enforce unfair and deceptive contracts.)

More On Possibility:

Objections to an overly broad view of possibility

Our world cannot possibly be anything but what it is since for every possible world, our world is what is actually our world.

Possible worlds other than ours (actual) are completely irrelevant and unobservable.  Minimization of unexplainable requires their nonexistence.

If something actually happens, it necessarily actually happens since for every possible world it happens in the actual world.

Exists means is.  If there is a possible world is which A is present, then there exists a possible world in which A exists.  By transitivity of existence in something, A exists.

Personal identity at any given time implies numerical identity.  Anything that is not in our world cannot be numerically identical to you since you are in this world and it is not.  At this moment, it is impossible for you to be anything but what you are since for every possible world, you exist only in this world, and since for every possible world, what happens in this world is what actually happens.

Let symbol □ denote 'necessarily', and ◊--possibly. false→□false and true→□true.  Under ordinary notation,  variables can be replaced with their values to evaluate truthfulness of statements.  Thus, for every A such that A is true or false, A→□A.  Possibility is useful only when the value is a function:  Let x be a variable denoting any  bird.  Let B(x) = x can fly.   Then ◊B and ◊(not B)  since some birds fly and some cannot.  However, the value of B is a function and thus is neither true nor false; "B→□B" is as meaningless as "72→□B".  Now, let P be "A person named George W. Bush is the president of the USA".  It is universally agreed that P is true or false (it is always assumed that P is about our world, so the assumption is implicit in the language), so P→□P.  Let P1(w)="In world w, a  person named George W. Bush is the president of the USA in that world". P1 is not a statement, so conceivably ◊ P1 and ◊(not  P1); however, forall w (P(w)→□P(w)).  Since we care and can change only our world, we care about truthfulness and possibility things such as B, P, and P1 (our world) but not P1.

In other words, possibility is a property of predicates, not statements.

Determinism

The argument for determinism is:

For any event P, let F(P) be "P will actually happen".

From what the discussion above, forall P (F(P)→□F(P)) and forall P (~F(P)→□~F(P)).  ('~' is a symbol for 'not'.)

Since forall P (F(P) or ~F(P)),   forall P (□F(P) or ~□F(P)),

which means that ~thereexists P (◊F(P) and ◊~F(P)),

which is implies determinism since indeterminism requires that for some P, it is possible that P will actually happen and it is possible that P will not actually happen.

Can you find a flaw (if any) in the argument?

Dmytro Taranovsky

November 8, 2002

Determinism Resolved

You exist and have feelings.  Your knowledge of reality is derived from your feelings.  If something exists, then its existence should be subject to an explanation.  Minimization of unexplainable requires that if there can be no reason (or evidence) for something to exist, then it does not exist.  You cannot see, hear, touch, feel, or smell the future or the past.  You have no evidence that the future or the past exists.  The future and the past are convenient in scientific models; however, truthfulness of science is limited to the observable predictions that it makes, and everything observed is in the present.  Our feelings universally tell us that the world changes.   Your thoughts change.  Buildings become created (become existing) and destroyed (cease to exist).  Moreover, causation, as opposed to correlation, is meaningful only in the context of change.  Accordingly, the most plausible theory is that the present is everything that exists.  The past existed but does not exist now.  The future will exist but does not exist now.  The present changes.  Free will is the ability to genuinely choose how to change the present.

Ordinarily, you cannot refer to nonexistent objects as though they exist.  For an ordinary reference to be valid, it is necessary that the object referenced either existed, exists, or will exist and is determined.  References to undetermined things are ambiguous and hence invalid:   If you reference the number x where x is 2 or 3 without specifying which, x is ambiguous, so the reference is invalid.  The actual future does not exist and have never existed.  Since it is not determined, it cannot be logically referenced.  The argument for logical determinism references the actual future and is therefore void.  What exists are the possibilities for the future:  The possibility that event A will happen and the possibility that event A will not happen.  You can reference the possibilities for the future.  You can refer to the future as a function of the possibility chosen.  In addition, you can reference an imaginary or semi-actual probability distribution of the possibilities for the future.  You can reference an imaginary future and even argue about the accuracy of that future.  A claim about the actual future, however, is like a claim about an undefined word.  It becomes meaningful only when the time comes or the word is defined.

Your choices are (or should be) based on the semi-actual probability distribution of possibilities for the future.  The semi-actual probability distribution is the best possible distribution based on the present and the past; for present and past events, the distribution is fully accurate.  (Because of free will, the actual probability distribution does not exist.)  Minimization of unexplainable is used to approximate that distribution.  Based on the distribution, the most probable future is defined.  Whenever you say that event A will actually happen, you mean that A will happen in the most probable future.  Let the future denote the most probable future.  Through free will, you can change the future.

Although the truth values of statements that refer only to unchanging objects (such as the specific objects in the world of human imagination and what happened at a specific time in the past) are constant, the truthfulness of statements like "No major earthquakes occurred on Earth in the last hour" can change because the objects referenced change.  Thus, if yersterday you stated "Event A will happen in 2010" and your statement was true yersterday, it is still possible that the future had changed, so as to make the statement false today.  There is no actual unchanging future, but there is a changeable future based on which you decide what to do and how to create the future.

It is natural and useful for human mind to simplify things.  Determinism, which is the idea that the future does not change, is just such an oversimplification.  The idea of determinusm can be useful, but it is fatalistic if accepted unconditionally.