In the previous post, I discussed modal logic and presented a paradox. This paradox runs deeper than I first thought, and in tonight's post I wish to discuss this paradox again from a different angle, from a perspective that brings to the surface this profounder aporia. I know the last couple of posts have been a little drier than my readers probably like, but I've been in the mood in recent months to discuss abstract ideas, to try to get my ideas down 'on paper' as best I can. Even though, stylistically, these posts lack the fireworks you find in quality literature, may seem a little boring on the surface, underneath there is a deeper wisdom, if this is the right word, than you find in more meretricious opinion pieces.
The paradox I presented in the previous post runs as follows: The proposition "It is necessary that p" implies "It is possible that p." The proposition "It is possible that p" implies "It is possible that not-p." Therefore the proposition "It is necessary that p" implies "It is possible that not-p" – a contradiction. In the previous post, I suggested that the way to dissolve this paradox is to presume that all propositions are either necessary or possible, but cannot be both. This attitude towards proposition entails a particular attitude towards the world, a picture of the world that has attracted many philosophers for a long time. Bertrand Russell and the early Wittgenstein both regarded the world as consisting of facts and believed that every true proposition corresponded with a particular fact in the world. Although both Russell and, I think, Wittgenstein disapproved of and dismissed modal logic, we can elaborate on the picture they embraced by supposing that all facts are either necessary or contingent. Every fact is either one or the other. Furthermore, a fact can be necessary, or it can be contingent, but it can't be both. This picture is the one we will be working with.
If we divide all facts into those which are necessary and those which are contingent, if we divide all true propositions into those which are necessary and those which are only possible, it seems, at first, that we have dissolved the paradox. The conjunct of "It is possible that p" and "It is necessary that p" is logically false. It is like saying, "The ball is blue, and the ball is red"; necessity and contingency are mutually exclusive properties. Only one of the two can be possessed by an preposition. Logically, if "It is necessary that p" is true "It is possible that p" is false, and the argument which led to the contradiction I described in previous paragraph is shown to be invalid. And so it seems we have dissolved the paradox.
Or have we? If the conjunct described above is false, it must be because one of the component propositions is false. Let us assume that "It is necessary that p" is true and "It is possible that p" is false. If "It is possible that p" is false, this implies "It is necessary that not-p" is true. Therefore "It is necessary that p" implies "It is necessary that not-p". So it appears we have arrived at another contradiction. If we assume, however, that "It is necessary that p" is false and "It is possible that p" is true, no contradiction arises. This applies, of course, to all propositions. In other words, then, there are no necessary propositions and no necessary facts. A fact can only be contingent and all true propositions express possibility rather than necessity.
I have been thinking about this issue a lot over the last week. When the idea of the second paradox, the other horn of the dilemma, occurred to me, it seemed at first that there were only two possible solutions: we could presume that all facts are necessary and that all true propositions express necessary truths, or we could presume that all facts are contingent and that all true propositions express only possible truths. A world which is a mixture of necessary and contingent facts leads only to paradoxes of the sort I've described. At first, I leaned towards the first option – in the previous post, I stated that I believed that all truths are necessary, that fate exists and free will doesn't. In this blog, in the posts about free will, quantum theory, and Donnie Darko, I have expressed the view that the world is deterministic, that all effects follow ineluctably from causes. But when I considered these paradoxes more rationally I realised that the only logical answer to them is to assume that all the facts that make up the world are contingent, and that all propositions can only express possibilities rather than necessities.
If this argument is correct, then there is no such thing as necessity. It might be helpful to enliven this discussion with some examples. The proposition, "It is necessary that Donald Trump is President of the United States" is false but the proposition "It is possible that Donald Trump is President of the United States" is true. The proposition "All cats are necessarily carnivorous" is false but the proposition "All cats are possibly carnivorous" is true. The proposition "It is necessary that the water molecule is dihydrogen monoxide" is false but the proposition "It is possible that the water molecule is dihydogen monoxide" is true. The proposition "The Morning Star is necessarily the Evening Star" is false but the proposition "It is possible that Morning Star is the Evening Star" is true. All of these propositions imply that their opposites might be true. Trump might not the President of the USA, all cats might not be carnivorous, the Morning Star might not be identical with the Evening Star, and so on. In order to resolve the paradoxes I described, we need to abandon our certainty that truths are knowable, embrace our uncertainty.
The only way out of the two paradoxes I have proposed is to suppose that all our knowledge is provisional.
The type of argument I am making is similar in method but opposite in conclusion to arguments made by Aristotle and by Richard Taylor. These two philosophers, one ancient and one modern, argued that fate exists. Consider the statement, "The First World War ended on 11 November 1918". If this is true, they argued, it was necessarily true. Currently, the US Congress has just begun the process of trying to impeach Trump. Suppose I say, "Sometime in the next three months, Trump will be impeached." This statement is, today, according to Aristotle and Taylor, either true or false: either way it implies that future events, like past events, are necessary, are in a sense preordained, unescapable, fated. They are necessarily true. One well known wannabe philosopher tackled this problem and sought to show that the fatalists were wrong. In an essay written in 1985, David Foster Wallace argued that propositions are necessarily true at some times and only possibly true at other times. It is a long while since I have read Time, Fate, and Language: An Essay on Free Will (I recommend it to readers of my blog) but it strikes me that Wallace could have simplified his argument if he had considered the idea that the notion of necessity, in modal logic, is incoherent, as I have argued in this and the previous post.
I'll summarise my argument in the following way. The conjunct made up of the two proposition "It is necessary that p" and "It is possible that P" is false. The conjunct made up of the two propositions "It is necessary that p" and "It is not possible that p" is false. Therefore, by the law of bivalence, the first proposition in each conjunct, "It is necessary that p", must be false. Therefore there is no such thing as a necessary proposition. All facts are contingent and all propositions are only provisional, possible.
This conclusion is, I concede, inconsistent with the conclusions I have drawn in some of my other posts. Can we reconcile fatalism with the contingency I have suggested is an inescapable feature of the way we use language? In the second paragraph, I limned a picture of the world, the picture presented by Russell and others: the world consists of facts, and true propositions correspond to particular facts. Perhaps this picture is wrong. Perhaps all facts are necessary, and all propositions are contingent. Propositions do not correspond to facts but rather betoken beliefs that we have acquired from others or inferred from observations of the world. Perhaps this is a better picture. We are licensed to say, for instance, "As far as I know, Donald Trump is President of the United States" or "As far as I know, all cats are carnivorous" or "As far as I know, the Morning Star is the Evening Star". I am unsure if this different picture, different perspective, can fully resolve the paradoxes I have discussed in this post and the previous post. But it might. The view of truth I have been trying to present in the last four posts is difficult for me to articulate clearly, but if any of my readers are familiar with the work of Richard Rorty, you might find that he has done the job for me. I hope so anyway.
Wednesday, 25 September 2019
Thursday, 19 September 2019
Modal Logic
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.
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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