In today's post, I wish to talk about modal logic, in particular alethic propositions. This post follows on from the previous two and will also refer to the two posts I wrote about quantum physics, "Probability and Schrodinger's Cat" and "Probability and Schrodinger's Cat Part 2". I don't have a clear idea of the direction this post will take; instead I think I'll work out what I want to say as I go along.
The major pioneer of modal logic in modern times is a chap called Kripke. When he was only nineteen and still a sophomore, he came up with a theory that has enormously influenced logic ever since. Kripke's contribution to philosophy was to say that the operators "It is possible..." and "It is necessary..." could be prefixed to propositional statements. I could say, for instance, "It is possible that my friend Harold is a lifeguard" or "It is necessarily the case that Germany lost the Second World War". In general, if a proposition such as "It is possible that P" is true, the proposition "It is necessary that not-P" is false, and if a proposition such as "It is necessary that P" is true, the proposition "It is possible that not-P" is false. In my view, Kripke's system is meretricious. It seems simple and obvious but in fact is neither. My aim in this post is to demonstrate that modal logic leads to paradoxes unless we recognise that probabilist propositions exist only in the minds of observers.
A simple example of a modal proposition is "It is possible that there is life on other planets". Another modal proposition is "It is possible that Mark Lundy didn't murder his wife and daughter." A third is the proposition "Magnetic monopoles exist". I imagine most of my readers would regard these three sentences as reasonably unproblematic – the three propositions deal with questions that are live issues in the world today. The paradox emerges when we deal with statements that are generally recognised as patently apparent truths or falsehoods, with statements that everyone knows to be true or knows to be false. Suppose I assert the following: "As of September 2019, it is possible that Donald Trump is President of the United States." I imagine at least some of my readers will feel that there is something fishy about this statement. Donald Trump simply is the President of the United States. You might nevertheless have the intuition that, despite its fishiness, it must be true. After all, if Donald Trump is the President of the United States, it must be possible for him to be President. A proposition like this could be reformulated or re-expressed. We could say "Donald Trump is possibly the President of the United States" is equivalent to "Either Donald Trump is President of the United States, or he is not President of the United States." This reformulation is true a priori. The sentence "Either p or not-p" is true of all propositions, including ridiculous ones, by the law of bivalence. I could say "It is possible that the moon is made of blue cheese" and it would be true, if we interpreted propositions dealing with contingent facts in this way. So the intuition that the Trump proposition must be true has been shown to lead to unreasonable and insupportable consequences.
Some of my readers may have the equally strong intuition that the fishiness of the Trump proposition I used as an example above suggests it must, somehow, be false. After all, if we say that it is possible that Trump might be the President of the United States, we are suggesting that it is also possible for him not to be President of the United States. And we know, empirically, that this must be incorrect. Because he is. We can approach this confusing and confounding situation from another direction. Consider the following two statements: 1. "It is possible that Donald Trump didn't become President in January 2017." 2. "It is possible that Donald Trump might not have become president in January 2017." From one angle it appears that these two sentences are equivalent. From another angle, they are quite different. The first appears to be saying something about the world, or about our knowledge of the world. The second is asserting that the world could have been some other way, is contingent, and hints at the idea, so dear to most people, that we all have free will and that there is no such thing as fate.
I am not a professional philosopher, have not read any recent literature about modal logic, but I suspect that the paradox I am going to attempt to articulate as clearly as I can in a moment is at once obvious and at the same time not one modal logicians have recognised, even though it is at the heart of the systems they embrace. Modal logicians know that the proposition "It is possible that p" does not imply "It is necessary that p" but I suspect that many would defend the converse relationship, saying that "It is necessary that p" implies "It is possible that p". They would argue that the set of all necessary truths is a subset of the set of all possible truths. The paradox is as follows. "It is necessary that p" implies "It is possible that p". "It is possible that p" implies "It is possible that not-p". Therefore, "It is necessary that p" implies "It is possible that not-p". We have arrived at a contradiction. The only way out is to suppose that the first inference, the first move, was wrong – if the proposition "It is necessary that p" is true, the proposition "It is possible that p" must be false. The set of all necessary truths and the set of all possible truths are mutually exclusive. A proposition can be necessarily true or possibly true, but cannot be both.
The issue I am engaging with indirectly is once again the issue of free will. Do people have it or don't they? I am inclined to believe that people don't have free will and that Fate exists. I believe that all truths are necessary and that the notion of possible truths and possible worlds is incoherent. Interestingly, the issue of whether people have free will or not is an issue that divides philosophers. Most academics working in Philosophy departments, apparently, are compatibilists – they believe it is possible to reconcile determinism with free will. Chomsky thinks we have free will but lack the cognitive firepower to make sense of the concept of it. Daniel Dennett and Rupert Sheldrake, despite being polar opposites in almost every other way, both believe in free will. Sam Harris doesn't. I sometimes disagree with Harris about political issues but I happen to agree with him on this philosophic bone of contention. For an interesting discussion of issues relating to free will and counterfactuals, have a listen to his conversation with Judea Pearl, available through Harris's podcast or on Youtube.
If all truths are necessary and the notion of a possible truth is incoherent, why then do human beings employ words like "possible", "improbable", "likely" and "unlikely" in their vocabulary? To answer this question and to resolve the paradox I submitted above, I am going to tell a small story. In mid 2015, after Donald Trump throws his hat into the ring and begins seeking after the Republican nomination, Jack and Jill go to live in the wilds of Borneo to study orangutans. They do not read the newspaper or watch TV for four years, and have no contact with the outside world at all. Yesterday, Jack says to Jill, "It is possible that Donald Trump might be President of the United States." Given this context, the fact that neither Jack nor Jill knows, this proposition no longer seems so absurd, so fishy. What Jack is saying is "As far as we know, Donald Trump might be President of the United States." It is posited as a possibility because neither Jack nor Jill has all the relevant information. This is the heart of my argument. All probabilistic statements, in my view, presume this caveat. As far as the astrophysicists and biologists know, there might be life on other planets. As far as the New Zealand body politic knows, Mark Lundy might be innocent. As far as the physicists who embrace the theories of Paul Dirac know, magnetic monopoles might exist. All estimates of probability, even an estimate as crude as the distinction between "possible" and "necessary" are, in a sense subjective, are based on models of the world that do not include all the information and relevant rules. Probabilistic statements arise from ignorance. If we could see the world in a purely objective way, if we could cleanse the doors of perception, if we were omniscient, all truths would be necessary. This is the idea that I was trying to express in the two posts on quantum physics, "Probability and Schrodinger's Cat" and "Probability and Schrodinger's Cat Part 2". I think these two posts are perhaps the most important I have written.
To try to sum up – a truth is possible if the people in the relevant linguistic community are in doubt as to whether it is true or not. A truth is necessary if everyone in the community is sure that it is true.
I hope the reader gets the points I'm trying to make. If I am unclear or have not persuaded you, I shall try to mount further arguments in favour of my position in the future. I hope, also, that you see the connection between this post and the previous two.
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