3 Nov 2015

Priest, Ch6 of Logic: A Very Short Introduction, “Necessity and Possibility: What Will be Must be?”, summary

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[Bracketed commentary and boldface (unless otherwise indicated) are my own. Please forgive my typos, as proofreading is incomplete. And since I probably make even bigger errors, it is best to work with Priest’s text directly, which I very highly recommend.]

Summary of

Graham Priest

Logic: A Very Short Introduction

Ch.6
Necessity and Possibility: What Will be Must be?

Very Brief Summary:
Modal operators allow us to say of a statement p that it is possibly true, ⋄p, or that it is necessarily true, ◻p. These modal operators are not truth-functional, since they do not alter the truth values in a mechanically consistent way. They can be understood using the notion of possible worlds, which can be diagramed using boxes. For a given world or situation, our own for example, we list certain states of affairs and facts along with their proper truth values. In the other boxes, we list those values as they are for the alternate worlds. If in all the worlds a certain fact is true, then it is necessarily true for any one of them. But if in at least one world it is not true, then it is only possibly true in some world where it is true. Thus, even if something is true in a world, that does not mean it is necessarily true; for, it could be otherwise. However, facts that the laws of physics prevent from being otherwise, no matter what variations are found in the other possible worlds, are necessary. Modal logic helps us see why Aristotle’s fatalist argument, that whatever happens cannot be otherwise, is fallacious. However, we might be able to formulate a stronger fatalist argument which says that true statements about the future are necessarily true, because they were true in the past, and all past facts are irrevocably true.

Brief Summary:
We can modify a statement of fact to indicate whether or not the referenced state of affairs is possibly the case or necessarily so. Modal logic allows us to deal with these modifications formally. Suppose “it will rain” is p. We write, “Possibly it will rain” as ⋄p, and we write “necessarily it will rain” as ◻p. Unlike truth-functional operators (like negation and conjunction), these modal operators do not alter the truth values of statements in a mechanically consistent way. To formally examine modally modified sentences, we think of there being other possible worlds about which we may make the same statements of fact, and these statements may be true or false depending on which alternate possible world it is in. In one possible world, it does rain tomorrow. But in another, it will not. We say something is possible when in at least one other world this state of affairs is false. However, no matter what possible world we conceive of, in all of them, if it rains, then fluid is falling. Such things which cannot be otherwise, when for example they are governed by fixed laws of physics, are considered necessarily true; for, in every other possible world they are true. We can diagram these possible world situations using boxes. In one box we give the statements of fact and their truth values for one situation or world (this world for example), and in other boxes we give the statements and their values for the other possible worlds. This helps us see which statements are necessarily true or false in one world and which are possibly so. This manner of formulation helps with certain debates, for example, it allows us to see that Aristotle’s argument for fatalism is fallacious. The argument makes us think that there is nothing we can do now to change the future, and also, that there is nothing in the past that we can regret or feel responsible for. The reasoning is as follows. If it is true that something will happen, then it will happen no matter what. But if it is false that something will happen, it will fail to happen no matter what. Either way, whatever happens occurs no matter what. By formulating this using modal logic, we see that it infers something incorrectly. There is a difference between the following two claims: 1) it is necessarily the case that if it is true that tomorrow I will get in an accident, then I will get in an accident, and 2) if it is true that if I will get in an accident, then I will necessarily get in an accident. If we just look at the semantic references, both formulations seem to have the same meaning. But on the level of their logical structure they are making different claims, and also structurally the second claim cannot be derived from the first, which is what is needed for the argument to hold. Aristotle’s fatalist argument would want you to believe that in every possible world you will get in an accident tomorrow, which is not so. It even acknowledges that the opposite could happen. However, there is a way to twist this fatalist argument a bit to remove that fallacy, and we may wonder whether or not this modification provides a valid argument for fatalism. We first say that there is nothing we can do now to change the past. This implies that states of affairs in the past are irrevocably true and statements about those situations are necessarily true. Now, suppose we do get in an accident tomorrow. This means it is true now if we say that we will. Suppose further that we said it yesterday also. We can say now that in the past it was true that we will get in an accident tomorrow. This means that it is irrevocably true that in the past we will get into an accident, and thus it is necessarily true that we will.

Summary

[So far we have been dealing with declarations of facts as they are. We have not yet added to our declarations whether or not the situations they indicate are possibly the case or necessarily the case.]

We often claim not just that something is so, but that it must be so. We say: ‘It must be going to rain’, ‘It can’t fail to rain’, ‘Necessarily, it’s going to rain’. We also have many ways of saying that, though something may, in fact, not be the case, it could be. We say: ‘It could rain tomorrow’, ‘It is possible that it will rain tomorrow’, ‘It’s not impossible that it will rain tomorrow’. If a is any sentence, logicians usually write the claim that a must be true as ◻a, and the claim that a could be true as ⋄a
(38)

[Recall from chapter 2 our discussion of truth functions. With these, we can decisively compute the values when operators are added to sentences. As we will see, the modal operators do not compute with mechanical consistency.] The modal operators ◻and ⋄ are not truth functions. Recall negation from chapter 2. If we know the truth value of a, then know that ¬a has the opposite value. [This is using the assumptions of classical logic. For computing values using non-classical logic assumptions, see chapter 5.] We can similarly compute the values for ab and a&b if we know already the truth values for a and b individually (38-39). It does not work this way for ⋄, since “you cannot infer the truth value of ⋄a simply from a knowledge of the truth value of a” (39). Priest gives the following examples. [He will take different sentences. One of them when adding ⋄ makes it false, and the other true. Thus the change it makes is not mechanically consistent for the same input value. This holds as well for ◻. He again will take two sentences, and show that when you add ◻ to one, it becomes false, and the other becomes true.] We take two sentences:

r: I will rise before 7 a.m. tomorrow.
j: I will jump out of bed and hover 2m above the ground.

Priest says that for him, both of these sentences are false. Were we to apply the same truth-functional operator to them, like negation, they should both then have the same output value. However, what happens when we add “It is possible that,” in other words, the ⋄ operator, to both of them?

r: It is possible that I will rise before 7 a.m. tomorrow.
j: It is possible that I will jump out of bed and hover 2m above the ground.

As we can see, it is true that the first one is possible, since “I could set my alarm clock and rise early.” The second one, however, is not possible, since it goes against the laws of physics. The same modal operator when applied to these false sentences produced a different truth value in each case. Now consider these sentences.

r: I will rise before 8 a.m. tomorrow.
j: If I jump out of bed tomorrow morning, I will have moved.

Both of them are true, Priest says. Now what happens when we add “It is necessary that”, the ◻ operator, to both of them?

r: It is necessary that I will rise before 8 a.m. tomorrow.
j: It is necessary that if I jump out of bed tomorrow morning, I will have moved.

In the first one, it is false, since Priest also could stay in bed for one reason or another. But the second one is true, since by doing the one action (jumping), one has thereby already done the other (moving). So again, by adding the necessity modal operator, we do not get an output that is mechanically consistent for the input.

Priest says that modal operators are also at times very puzzling. His example is Aristotle’s argument for fatalism.

Priest explains, “Fatalism is the view that whatever happens must happen: it could not have been avoided” (39). Many find it appealing, since if something goes wrong in their lives, they will not feel guilty for having failed to prevent it. For, nothing that happens can be prevented anyway. It is just fate. But as Priest notes, fatalism “entails that I am powerless to alter what happens, and this seems plainly false” (39). For example, imagine that you were involved in a traffic accident today. [Granted, you cannot change the fact that someone else ran through a red light in front of you, leaving you with no time to stop before hitting them, for example. However,] you could have avoided the accident situation altogether merely by having taken one of the many other routes [or by leaving at even a slightly different time, and so on.] Aristotle is making a different point. [Priest places some text in boldface, because it will become important later. So in this case, the boldface is not mine.]

Take any claim you like – say, for the sake of illustration, that I will be involved in a traffic accident tomorrow. Now, we may not know yet whether or not this is true, but we know that either I will be involved in an accident or I won’t. Suppose the first of these. Then, as a matter of fact, I will be involved in a traffic accident. And if it is true to say that I will be involved in an accident then it cannot fail to be the case that I will be involved. That is, it must be the case that I will be involved. Suppose, on the other hand, that I will not, as a matter of fact, be involved in a traffic accident tomorrow. Then it is true to say that I will not be involved in an accident; and if this is so, it cannot fail to be the case that I won’t be in an accident. That is, it must be the case that I am not involved in an accident. Whichever of these two does happen, then, it must happen. This is fatalism.
(41)

Priest will now examine these ideas using “a standard modern understanding of modal operators” (41). [My understanding of the following is a bit flawed. I do not understand the difference (if there is one) between a world and a situation. It is not simply that a situation is found in a world, since a situation that can result causally from another situation is itself called a possible world (rather than being just another situation found within a shared world or in another world of events/situations). I also do not know exactly what is meant by “associated with” in the formulations that speak of possible worlds that are associated with certain situations. I will offer my best guess-explanation here, and then I will quote. We have propositions that may describe states of affairs. We may name the propositions with letters, like a and b. Depending on what is happening in reality, a and b can be true or false. This is determined on the basis of whether or not the proposition corresponds to the real states of affairs they refer to. A situation is a collection of such facts/statements. And, a situation can also be called a world. Different world/situations can have the same propositions describing states of affairs that may or may not hold in that world/situation. But what makes these different world/situations distinct from one another is that in any of them at least one of the facts will have a different truth value. Worlds for which the same sets of statements can be made are “associated with” each other, and they are “possible worlds” with respect to one another. The facts within an associated worlds need not all take up the same temporal location. So a may be a statement regarding a present event, and b about a later one. We then look at all the different assignments of truth values for situation/worlds with the same statements of fact, and we speak with regard to one particular one (our own perhaps). If a statement is true in all the other situation/worlds, it is necessarily true in the one in question. If it is a true statement in only some or one of them, then it is possibly true in the one in question.]

We suppose that every situation, s, comes furnished with a bunch of possibilities, that is, situations that are possible as far as s goes – to be definite, let us say situations that could arise without violating the laws of physics. Thus, if s is the situation that I am presently in (being in Australia), my being in London in a week’s time is a possible situation; whilst my being on Alpha Centauri (over four light-years away) is not. Following the 17th-century philosopher and logician Leibniz, logicians often call these possible situations, colourfully, possible worlds. Now, to say that ⋄a (it is possibly the case that a) is true in s, is just to say that a is in fact true in at least one of the possible worlds associated with s. And to say that ◻a (it is necessarily the case that a) is true in s, is just to say that a is true in all the possible worlds associated with s. This is why ◻ and ⋄ are not truth functions. For a and b may have the same truth value in s, say F, but may have different truth values in the worlds associated with s. For example, a may be true in one of them (say, s′), but b may be true in none, like this:
(Priest 41)

(diagram based on Priest 42a)

Priest then will show how this means of describing possible worlds allows us to analyze inferences that use modal operators. In particular, we will see why the following inference is invalid:

a   ⋄b
⋄(a & b)

To see why it is invalid, we now suppose two other situations associated with s,  namely s1 and s2. The three situations have the following truth values.

(based on Priest 42c)

[Even though a is true in some world, and b is true in another, there is no world in which both are true. Thus it is not possible that their conjunction is true, and the inference then is not valid. There is something that I find interesting in the following reasoning. a is false in s. But it is true in s1. This means that even though it is false in s, it is still possible in s. The reasoning seems to be that since there is another world “associated with” (having states of affairs that are describable by the same propositions, regardless of the truth value of those propositions) s where a is true, that makes it possible even within s. For, if it happens elsewhere in a similar world, then it could happen in this world. And, if in our world and in  no world is a, then it must be necessarily false. For, it cannot possibly happen regardless of whatever legitimate contingent variations we place on our own world. So something is possible in our world if it could be otherwise (and in fact is otherwise in a possible variation of our world), and something is necessary if it could not be otherwise (and in fact is not otherwise in any possible variation of our world). So one way of looking at possibility is to think, ‘could it happen in our world’? But with this modal logic view, we think, ‘is it in fact happening in another world similar to ours?’]

a is T at s1; hence, ⋄a is true in s. Similarly, b is true in s2; hence, ⋄b is true in s. But a&b is true in no associated world; hence, ⋄(a&b) is not true in s.
(42d)

We then look at the following inference:

a   ◻b
◻(a & b)

It is valid. But why? [The basic reasoning seems to be that since a and b are true in all worlds, then they must as well always be true together in those worlds. For, there would be no world where one is true and the other not.]

if the premisses are true in a situation s, then a and b are true in all the worlds associated with s. But then a&b is true in all those worlds. That is ◻(a&b) true in s.
(43)

Recall our examination of Aristotle’s fatalism. We still have one more thing to discuss before we return to it, namely, the logical operator called the conditional, →. “If a then b” would be written as ab. There is an important inference that uses the conditional. [It seems basically to say that if we know a, and if we know that a implies b, then we also know that b.]

a   ab
b

Priest gives this example: “If she works out regularly then she is fit. She does work out regularly; so she is fit” (43). We still use the medieval name for this inference: modus ponens, which translates literally as “the method of positing” (43).

In order to apply all this to Aristotle’s fatalist argument, we need to look at a formulation that is relevant to the reasoning that he is using:

if a then it cannot fail to be the case that b.

Priest says that sentences like this are ambiguous. Let us look at the two meanings. Meaning 1) “if a is, as a matter of fact, true, then b is necessarily true. That is, if a is true in the situation we are talking about, s, then b is true in all the possible situations associated with s. We can write this as a→◻b” (43d). An example of this reasoning would be thinking the following: “You can’t change the past. If something is true of the past, it cannot fail to be true. There is nothing you can do to make it otherwise: it’s irrevocable” (44a).

Meaning 2) “if a then it cannot fail to be the case that b,” which would be written as: ◻(ab). This is a very different meaning than the first one. [The difference seems to be on where the necessity is found. In the first meaning, the necessity is on b being true, if a is true. In the second meaning, the necessity seems to be on the implicatory relation between the two. So for the first, we might say, “it cannot be otherwise that if you have a, then you have b.” We cannot for example not have b. For the second, we might say “it cannot be otherwise that a implies b”. We cannot for example have that it is not the case that a implies b. He gives an illustration to clarify the difference. For the second case of ◻(ab), we could  not that if a person is getting a divorce, they must first already be married. This means that in no other world can you have one without the other. But, regarding first case of a→◻b, if a person gets a divorce, that does not mean that they are married and must stay married forever. I am not sure how this applies to possible worlds. It would seem to be similar to the example of being in Sydney and going to either London or Alpha Centauri in a week’s time. In one possible world, Priest has gone to London. But in no possible world has he gone to Alpha Centauri in that time, because it is physically impossible. Thus, the “possible worlds” can be understood as future variations of a given world. In no world - past, present, or future – can you get a divorce without first being married. This exemplifies the reasoning for ◻(ab). But this is different than the following claim. Supposing that a person really is going to get a divorce, then in no possible world does that person just stay married. This exemplifies the reasoning for a→◻b.]

The second meaning of a conditional of the form ‘If a then it cannot fail to be the case that b’, is quite different. We often use this form of words to express the fact that b follows from a. We would be using the sentence like this if we said something like ‘If Fred is going to be divorced then he cannot fail to be married’. We are not saying that if Fred is going to be divorced, his marriage is irrevocable. We are saying that you can’t get a divorce unless you are married. There is no possible situation in which you have the one, but not the other. That is, in any possible situation, if one is true, so is the other. That is, ◻(ab) is true.

Now, a→◻b and ◻(ab) mean quite different things. And certainly, the first does not follow from the second. The mere fact that ab is true in every situation associated with s, does not mean that a→◻b is true in s. a could be true in s, whilst ◻b is not: both b and a may fail to be true in some associated world. Or, to give a concrete counter-example: it is necessarily true that if John is getting a divorce, he is married; but it is certainly not true that if John is getting a divorce he is necessarily (irrevocably) married.
(44)

[I am not sure I completely grasp the difference yet. One important idea seems to be the following. We have a which can be true or false in any of the associated worlds. And the same holds for b. Now, if we look at the truth table for implication (ab), we see that it is only false if the a is true and the b is false. That means, the implication is true if both a and b are false and also if a is false and b is true (and as well of course if they are both true).

It seems also that Priest describes the following scenario. In a given world/situation, a is true and b is true, and hence ab is true. But in another possible world, a is false and b is false, thereby making their implication also true.

This means that in all the worlds, ab is true, and yet in one world, b is false. Thus in the world where b is true, it is not necessarily true, since there is an associated world where it is false. In other words, ◻(ab) does not mean a→◻b.]

Priest now returns to Aristotle’s fatalist argument. Recall the boldface sentence from a prior quotation: “if it is true to say that I will be involved in an accident then it cannot fail to be the case that I will be involved.” This sentence takes the form “if a then it cannot fail to be the case that b”, which ambiguously can mean either a→◻b or ◻(ab). In fact, the argument exploits this ambiguity. We will make a be the sentence “It is true to say that I will be involved in a traffic accident;” and b will be “I will be involved (in a traffic accident)” (44d). This means that the sentence: “if it is true to say that I will be involved in an accident then it cannot fail to be the case that I will be involved” is true in the sense of ◻(ab). [For, there could not be a world where you will be involved in a traffic accident but will also not be involved. But this is not yet fatalism, which would say that nothing else could have happened but that you get into an accident. To arrive at the fatalist argument that whatever happens could not have been otherwise,] we still need, however, to establish a→◻b. This, we cannot do. [In the following, I include bracketed material for clarity.]

After all, the next step of the argument is precisely to infer ◻b from a by modus ponens. But as we have seen, 2 [a→◻b] does not follow from 1 [◻(ab)] at all. Hence, Aristotle’s argument is invalid. For good measure, exactly the same problem arises in the second part of the argument, with the conditional ‘if it is true to say that I will be involved in an accident then it cannot fail to be the case that I won’t be involved in an accident’.
(45, bracketed text my inclusion)

[To recall, here again is the second part of the argument: “Suppose, on the other hand, that I will not, as a matter of fact, be involved in a traffic accident tomorrow. Then it is true to say that I will not be involved in an accident; and if this is so, it cannot fail to be the case that I won’t be in an accident. That is, it must be the case that I am not involved in an accident.”]

Priest then notes that there is a similar argument which cannot be answered as easily as we did with Aristotle’s. Consider statements about a past fact. If they are true, it would seem to mean that they are necessarily true [for, the past cannot be changed. I do not have a firm grip on why, but my guess is that it has to do with the conditions of physical possibility. It is not physically possible for events now to change situations in the past. Therefore, true statements about the past are irrevocably and thus necessarily true. For, in no possible world can we have a statement about a past fact be true in the past but later become false in forthcoming present. In other words, since there is no retroactive revision of reality’s facts, we are bound to say that all past facts are necessary.]

It is impossible now, to render it false. The Battle of Hastings was fought in 1066, and there is now nothing that one can do to make it have been fought in 1067. Thus, if p is some statement about the past, p→◻p.
(45)

Now consider some statement about the future. Again, for example, let it be the claim that I will be involved in a traffic accident tomorrow. Suppose this is true. Then if someone uttered this sentence 100 years ago, they spoke truly. And even if no one actually uttered it, if anyone had uttered it, they would have spoken truly. Thus, that I will be involved in a traffic accident tomorrow was true 100 years ago. This statement (p) is certainly a statement about the past, and so, since true, necessarily true (◻p). So it must necessarily be true that I will be involved in a traffic accident tomorrow. But that was just an example; the same reasoning could be applied to anything. Thus, anything that | happens, must happen. This argument for fatalism does not commit the same fallacy (that is, use the same invalid argument) as the first one that I gave. So is fatalism true after all?
(45-46)

[The following is quotation.]

Main Ideas of the Chapter

● Each situation comes with a collection of associated possible situations.

● ◻a is true in a situation, s, if a is true in every situation associated with s.

● ⋄a is true in a situation, s, if a is true in some situation associated with s.
(quoted from Priest, 46, boldface his)

From:

Priest, Graham. Logic: A Very Short Introduction. Oxford: Oxford University, 2000.