In the previous three posts, I have been talking about modal logic, and I wish to do so again today. As I've said before, this blog is a process rather than a destination, an opportunity to think out loud as it were – and I wish to amend some of the conclusions I came to in the previous posts. It is not so much that I have changed my mind, as that I have thought through the problems a little more clearly. It might be that many modern modal logicians, and Kripke himself, have come to similar conclusions to the conclusions I have reached and intend to spell out, but I need to be honest and say that I haven't actually read Kripke's Truth and Necessity and so am basing my views on perhaps an imperfect understanding of Kripke's work. I think it probable my view differs from his. Although I am not as well informed as others about his work, I still think what I have to say is interesting and perhaps novel, if the reader is interested in logic and philosophy, and can follow my argument.
We start with a picture of the world inspired by Bertrand Russell. The world consists of facts. We employ language to talk about the world, and every true proposition corresponds with a particular fact in the world. A fact can be either necessary or contingent. To say a fact is 'contingent' is to say that it could have been different, or that there are other possible worlds in which it is different. For instance, consider the sentence: "Donald Trump definitely became President in January 2017, but he might not have become President in January 2017". This sentence is ambiguous because English is ambiguous. From one perspective it seems to express something oxymoronic or paradoxical; from another perspective, it simply expresses the common-sense notion that the future is indeterminate and so past events could have happened otherwise. Donald Trump's past investiture is a contingent fact. Consider now the sentence "A water molecule is definitely H2O but it might not be H2O" or the sentence "All cats are carnivorous but some cats might not be carnivorous" or the sentence "All bachelors are unmarried but some bachelors might be married." These propositions we feel must be false, because they correspond (or, more exactly, fail to correspond) to facts that are necessary rather than contingent. A necessary fact is true in all possible worlds, whereas a contingent fact is true only in some possible worlds. We could add to the picture that all analytic truths are necessary, although not all necessary facts are analytic. Generally speaking, we tend to have a feeling about facts, a feeling that a particular fact is necessary or that it is contingent, but an exact determination of which category a fact falls into can be difficult to make. It is possible to argue that all facts are necessary and that there are no contingent facts at all; it is also possible, although far more unconventional, to argue that all facts are contingent and that there are no necessary facts. The problem of how we divide the world into necessary and contingent facts is extraordinarily difficult to solve but we don't need to solve it to deal with modal logic because, in truth, the issue of necessity vs. contingency is not the issue modal logic was invented to address. Modal logic is concerned with propositions rather than facts.
Modal logic, in the form Kripke first presented, proposes that we can prefix an alethic operator to propositions, an operator of the form "It is necessary that..." or "It is possible that..." I intend to argue that any proposition of the form p falls into one of three groups: "It is necessary that p", "It is necessary that not-p" or "It is possible that p". To assert of a proposition "It is possible that p" is also to assert "It is possible that not-p." This three-part partition reflects a person's knowledge of the world. If I say, "It is necessary that Donald Trump is President of the United States," I am saying, "I know with absolute certainty that Donald Trump is President of the United States." If I say, "It is necessary that Nancy Pelosi is not President of the United States," I am saying, "I know with absolute certainty that Nancy Pelosi is not President of the United States." If I say, "It is possible that life exists on Mars," I am saying "I don't know if life exists on Mars or not." Suppose you and I agree that some proposition p is true beyond a shadow of a doubt. We can then say to each other "It is necessary that p". If we agree with absolute confidence that p is false, we can say to each other "It is necessary that not-p." If we agree that neither of us know for sure if p is true or false, we can say to each other, "It is possible that p." (Of course, you and I might disagree about whether a proposition is necessarily true, necessarily false, or simply possible, but this type of disagreement is not something I wish to discuss in this post.) Modal logic does not tell us that the world itself is uncertain; rather, it tells us that a person's knowledge of the world is uncertain. A proposition expresses a belief about the world. All propositions, all beliefs, fall into one of the three groups, and no proposition can be in more than one of the three groups at the same time.
In the post "Modal Logic" I showed that a paradox seems to arise from the more conventional way of looking at modality. This paradox can be expressed 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" – a contradiction. In that post, I argued that we could dissolve the paradox by upholding as a foundational axiom the doctrine that a proposition can be necessary or possible but can't be both at the same time. However, in "Modal Logic Part 2", I showed that this strategy also results in a paradox. "It is necessary that p" implies "It is not possible that p" and "It is not possible that p" implies "It is necessary that not-p", therefore "It is necessary that p" implies "It is necessary that not-p"– another contradiction. In the same post I concluded that the only way to avoid both paradoxes is to say that there are no necessary propositions at all, that all propositions are probabilistic. In the post immediately preceding this one, "Logical Investigations", I showed that these paradoxes arise from six basic assumptions or premises about modal logic. Obviously, if we wish to retain the notion that there are necessary propositions as well as possible proposition, there must be something wrong with these six premises.
I want now to list four alternative premises or axioms, premises different from those I presented in "Logical Investigations". These premises harmonise with the picture of the world I sketched out earlier in this post and have the added advantage that neither of the two paradoxes I mentioned above follows from them. These four premises are:
1. "It is necessary that p" implies "It is not possible that p", "It is not possible that not-p" and "It is not necessary that not-p".
2. "It is possible that p" implies "It is not necessary that p", "It is not necessary that not-p" and "It is possible that not-p".
3. "It is not necessary that p" implies "It is possible that p" and "It is possible that not-p", or "It is necessary that not-p".
4. "It is not possible that p" implies "It is necessary that p" or "It is necessary that not-p", and "It is not possible that not-p".
These four premises provide a comprehensive alternative to the premises I presented in "Logical Investigations".
To reiterate, these four premises fit with the picture of the world I painted earlier in this post, and do not result in either of the paradoxes I have shown occur if we take a more conventional way of approaching modal logic. Moreover, this picture of the world aligns with the picture of the world I outlined in the posts "The analytic-synthetic distinction", "Fictional objects" and the two posts about quantum physics. This view, my view, is that reality is partial, subjective, and exists principally in the mind of conscious observers.
A curious extension of modal logic is second-order or second-level modal logic. In the picture I have drawn, all propositions fall into one of three camps: the necessarily true, the necessarily false, and the possible. However, you and I might be uncertain about which of the three categories a given proposition falls into. This uncertainty means that we can have sentences such as "It is possible that it is possible that p" and "It is possible that it is necessary that p". We could say "It is necessary that it is possible that p" – a type of sentence very appropriate to quantum physics. We could say that "It is possible that p" implies "It is necessarily possible that not-p". We could even say that "It is not possible that p" implies "It is possibly necessary that p." Obviously extending modal logic to encompass such meta-statements may result in things that look like paradoxes, but these paradoxes are less serious than the ones I have discussed earlier.
I hope this discussion is not so brief and curtailed that I have failed to make my case.
I'll finish this post by making an important point about my view of the world. It might seem like I endorse an extreme form of postmodernism, that I believe reality is created by conscious minds. A couple of years ago, I read "The Rhetoric of Fictionality" by Richard Walsh who appears to have this view– Walsh seems to subscribe to the doctrine that consciousness imposes form on "formless sense data". I myself do not believe this to be true though. I believe, rather, that regularities do indeed exist in the world and that we label these regularities with words. Two-hundred years ago, a Maori wandering in the bush would be able to distinguish between different types of tree – he or she could describe one tree as a Rata, another as a Kauri, another as a Totata, another as a Manuka. The natural world is divided into different types of object, different species of flora and fauna, and different people can agree that one type of object is different from another, even if we use different words to describe the same type of living being. This is a simple fact about the natural world. In the man-made world I live in, I can also distinguish between different types of object – between a sewer grate, a sign post, and a park bench, for instance. Whether we are dealing with a natural or man-made environment, different types of object present themselves to the senses. Humans, like other animals I believe, sort the world into different categories, but unlike other animals associate a given category with a spoken sound or written mark. The instinct to categorise things is innate. The problem arises when we take this cataloguing too far, by supposing, as psychiatrists do for instance, that human beings can be sorted into separate categories.
Once again I have found this post difficult to write. I feel tired and woolly-headed all the time which makes it difficult to express myself clearly. Still, I wanted to get my thoughts written down and I think I have achieved this.
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