allzermalmer

Truth suffers from too much analysis

Possibility and Necessity

Posted by allzermalmer on January 14, 2012

This blog is going to deal with logically necessary and logically possible. This is slightly different from Avicenna, but mostly based on Modal Logic, or some of the basic ideas of Modal Logic.

Because it is a form of logic, it also deals with one of the foundations of logic. Logic is concerned with statements, and the inferences that we draw from these statements. It is about having a couple of statements, and seeing if we can draw another statement with those statements that we accept. Logic helps give us some rules to follow in order to say that we drew a statement from the other statements that we held to, in a correct manner.

Law of Identity is “every individual thing is identical to itself”. Law of Excluded Middle is “every statement is either true or false”. Law of Non-Contradiction is “given any statement and its opposite, one is true and the other false”. And with possibility and necessary, they are mostly based on the Law of Non-Contradiction.

Possible means not self-contradictory. For example, “The sun won’t rise tomorrow” or “I ran a 2 minute mile” are possible. There’s nothing logically self-contradictory there. Necessary means self-contradictory to deny, which are based on logic, meaning of concepts, or necessary connection between properties. For example, “2+2=4” or “a bachelor is an unmarried male”.

A possible world is “a consistent and complete description of how things might have been or might in fact be.” A possible world is a consistent world, and this means that the statements that describe a possible world don’t entail self-contradiction. We can’t have statement X of possible world N and statement ~X of possible world N, being both affirmed at the same time like “statement X of possible world N and statement ~X of possible world N”. To do this would be to affirm a contradiction, or show that that possible world couldn’t exist because it’s a contradictory world. But the actual world is a description of how things in fact are. Yet, it would seem, that the actual world has to be consistent as well, which means there are no self-contradictions in the world. So the actual world is a possible world itself.

One of the difference between necessary and possible is that necessary statements are known to be true or false without experience. This means that information obtained from observation or sense-perception play no rule in determining if the necessary statement is true. This means we can know that a statement is true without recourse to evidence supplied by observation. But possible statements are known to be true or false with experience. This means that information from observation or sense-perception plays a part in determining if the possible statement is true. This means we know that a statement is true with recourse to evidence supplied by observation.

(A side note is that a necessary statement is a possible statement as well. This is because a necessary statement is a statement that it would be self-contradictory to deny, but the statement itself shouldn’t be self-contradictory and that’s what a possible statement is as well.)

From this idea of possible and necessary, we can say there are three types of statements. There are necessary statements, impossible statements, and contingent statements. A necessary statement is a statement that couldn’t be false. A impossible statement is a statement that couldn’t be true. A contingent statement is a statement that could be true or could be false, or could have been true or could have been false, or could be true in the future or could be false in the future.

So take “a bachelor is an unmarried male”. This is a necessary statement and means it’s necessarily true. But now say that I say “a bachelor isn’t an unmarried male”. This is a impossible statement, and means it’s necessarily false. To actually affirm the second statement is to affirm something that is false. Now say that I affirm “Justin Bieber is a bachelor”. That statement is a contingent statement. This means that Justin Bieber is a married male or isn’t a married male. The only way we could tell which of the two propositions is true is through experience.

So take the statement “Justin Bieber is a bachelor“, and accept it’s a contingent statement. So when we say “‘Justin Bieber is a bachelor‘ is a contingent statement”, we are also saying “‘Justin Bieber is a bachelor‘ is possible and not ‘Justin Bieber is a bachelor‘ is possible.”But take the same statement, and accept it’s a contingent truth. So when we say “‘Justin Bieber is a bachelor‘ is a contingent truth”, we are also saying “‘Justin Bieber is a bachelor’ is true but could have been false.”

As Raymond D. Bradley said, “Our own world – the real world, the actual one – is just one of many possible worlds. Indeed, it is just one of infinitely many, since for any possible world containing say n atoms there is another logically possible world containing n+1 atoms, and so on ad infinitum.” So there are an infinity of possible worlds, or possible ways of describing what might have been or what might be. Say we have the possible world of N, and within this world it contains M and M contains 140 particular things that are M. That is one possible world, but anther possible world, which is logically contradictory from N, and we can call it N*, contains 141 particulars in M. But 140 is logically contradictory from 141. So N is one possible world and N* is a different possible world, but they are logically contradictory from one another.We can continue on doing this infinitely, and so there’s an infinity of possible worlds.

Now science is concerned with contingent statements. This is because science is said to be empirical. The quotes, in order of the authors, are from Richard Feynman, Pierre Duhem, Stephen Hawking, and Henir Poincare: “The principle of science, the definition, almost, is the following: The test of all knowledge is experiment. Experiment is the sole judge of scientific “truth.””; “Agreement with experiment is the sole criterion of truth for a physical theory.”; “[the] scientific method…w[as]…developed with goal of experimental verification.”; “Experiment is the sole source of truth. It alone can teach us something new; it alone can give us certainty. These are two points that cannot be questioned.” Now experiment is based on coming to human sense-perception. This can be anything from looking at the squirrel climbing a tree to reading the numbers off of a volt-meter.

The reason that science wants to deal with contingent matters, besides dealing with sense-perception, is that science likes to try to have the ability to be shown that the theories are false. But if science only dealt with necessary statements, then scientific statements could never be shown to be false. For if a statement was presented that said a necessary statement was false, that statement would be an impossible statement and necessarily be false. But a contingent statement can be shown to be false.

But science creates models of how the actual world could possibly be. Take this example of what Richard Dawkins says of science and what science does: “There is a less familiar way in which a scientist can work out what is real when our five senses cannot detect it directly. This is through the use of a ‘model’ of what might be going on, which can be tested. We imagine- you might say we guess- what might be there. That is called the model. We then work out (often by doing a mathematical calculation) if the model were true. We then check whether that is what we see…We look carefully at the model and predict what we ought to see (hear, etc.) with our sense (with the aid of instruments, perhaps) if the model were correct. Then we look to see whether the predictions are right or wrong.  ”

So science creates a model of what is possible, and what happens in this possible world. And we can deduce what should be observed if this possible world is the actual world. We have the possible world of N. But N is our model. So, “If N then O. O. Therefore N.” This is the fallacy of affirming the consequent. Just because the model has a true prediction in the actual world, that doesn’t mean that the model is how the actual world is. In other words, just because this possible world (model) had one right prediction of how the actual world is (which is also another possible world), doesn’t mean that this actual world is that possible world. It also wouldn’t matter if the model has made nothing but correct predictions up till now, because the problem is still there.

If I robbed a bank, then I have 100 million dollars. I have 100 million dollars. Therefore, I robbed a bank. But me having 100 million dollars is also consistent with me winning the lottery, it is also consistent with me investing my money in a way where I got a lot of return in my investments, me starting up a business and my business got me 100 million dollars, or me getting the money from the death of a family member as a part of inheritance. In other words, there are many other possibilities that are consistent with the actual results or observations. As W.V. Quine once said, “Whatever observation would be counted for or against the one theory counts equally for or against another.”

This seems to raise a skeptical problem.

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