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.
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