An apology for modal logic.

Lucian Wischik. August 1996.

My brother doesn't believe in modal logic. This whimsical essay is an argument against him, in the style of Three dialogs between Hylas and Philonous by Bishop Berkely.

An Apology for Modal Logic

Lu.  Damon — hello. I have had some curious thoughts that I would like to discuss. Do you feel like a game of Dialectics? I will be Philonous, and you Hylas.

Damon.  That is not fair: I always have to play Hylas. You just have to use Philonous' clever argument techniques, and Hylas' common sense does not stand a chance. What if we swap this time? You be Hylas, and I will be Philonous. Ready? Good morning, Hylas: I did not expect to find you abroad so early.

Lu.  It is indeed something unusual; but my thoughts were so taken up with a subject I was discoursing of last night, that finding I could not sleep, I resolved to rise and take a turn in the garden.

Damon.  It happened well. Blah, blah, nature is beautiful, blah, but what were your thoughts, that caused you to wake up so early?

Lu.  I was talking to a mediaeval logician who claimed that there could be different modes in which a proposition could be true or false. He said that something might necessarily be true; or that it might possibly be true or false, or that it might necessarily be false. I think that this makes Sense, but he said that others had their doubts; and you must surely know by now, in what respect I hold Common Sense.

Damon.  Indeed yes, brother dearest. In this respect your common sense led you astray. If you had studied Maths at Trinity College, as I have done, it would be painfully obvious that the mediaeval logician was wrong. Let us use Measure Theory for this particular problem: To a statement whose validity we do not at the moment know, we can ascribe a probability for its truthfulness. When we aquire further information, then our estimation of its probability might shift to one or to zero. But, whatever our estimation, the actual value of the statement is fixed and defined to be either true or false.

Lu.  I think I understand...

Damon.  Since I studied Maths at Trinity College, and since you did not, I shall provide a simple example. Imagine a die that has been thrown, but that its results are kept from you. The result of this throw is called a random variable; and this random variable may take values that are elements of the result space, which in this case is the set of numbers from one to six. What probability would you ascribe to the random variable having the value, say, three?

Lu.  One-sixth, of course.

Damon.  That is correct. Now, I happen to know that the die landed on a 'six,' but I do not tell you. What probability would you ascribe now?

Lu.  It would have to remain one-sixth, since I do not actually know the result.

Damon.  Correct again. The actual value is six, and to say that it is necessarily six or possibly six or definitely six is meaningless. You can only talk about the value of the random variable, rather than it necessarily being one value or possibly another. Now, suppose you knew that this particular die was loaded by a nefarious younger brother, such that it always landed on a six. How would your estimation of its probability change?

Marcus.  I didn't do anything. Anyway, I haven't even seen the die before in my life, and I only needed it to play a quick game of Ludo.

Lu.  Marcus, you cannot butt in like that. This is a dialectic, so it has to be between two people. Philonous, my probability values would change now so I would estimate that it would land on six with a probability of one.

Damon.  Yes, you are correct. Your estimation of the probability has changed, but it is independent of the actual value of the random variable. In the same way, a proposition is a random variable which can take the value true or false but which cannot take the value necessarily-true or necessarily-false — since these simply are not members of the result-space. I should add that, when we take about your estimation of probability, we are assuming that you are reasoning correctly and that you are making no unjustified concLusions. This might not necessarily be true in your case, but we could just pretend that you had studied Maths at Trinity College.

Lu.  That makes Sense; and thank you.

Damon.  Now, suppose that a particular statement had a probability of one. By the very definition of probability, it must be true. If the statement had a probability of zero, then of necessity it would not be true. It does not matter that we are talking about the probability of an event that may or may not be in the past, since the notion of probability applies to information gained rather than to time.

Lu.  So, Philonous, you are saying that knowledge of the probability of a particular thing lets us come to concLusions about that thing?

Damon.  That is right. There is more you can say about probabilities. Suppose that you had deduced from your axioms that a particular statement was true. Then you could say that it was true with a probability of one. Similarly, if you said that a statement was true with probability one, then you could be certain that it is actually true. Further, if we are sure with probability of one that one event leads on to another, then whenever we are sure of the first event with probability one, we can be sure of the second with probability one.

Lu.  I see from what you say that how it is possible to talk about probabilities. Since I find all this talk about probabilities confusing, may I introduce a shortcut? Let us say that the statement "I can deduce that something is true," is the same as saying that my estimation of its probability is one. "I can deduce that something is false," is like saying that my estimation of its probability is zero; and "I cannot deduce either that something is true or that it is false," would be like saying that the probability is somewhere in between.

Damon.  Yes, this understanding of deduction is correct. Since you did not study Maths at Trinity College, perhaps I should write these things down for you to remember.

D p => p
D (p=>q) => (Dp => Dq)
:- r -> :- Dr

Lu.  Hold a minute, Philonous. Have you not just written down precisely the laws of Modal Logic, but replacing the word 'it is necessary that' with the words 'it can be deduced that' ?

Damon.  Do you know, Lu, perhaps in this case your Common Sense was right after all: Modal Logic, with its rules about necessary truths and falsehoods, does make perfect sense, just so long as we replace the phrase 'it is necessary that' with 'it can be deduced that.' I shall leave you for a little while so that I may learn about Modal Logic.