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by Corry Shores
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[Priest, Introduction to Non-Classical Logic, entry directory]
[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]
Summary of
Graham Priest
An Introduction to Non-Classical Logic: From If to Is
Part I:
Propositional Logic
7.
Many-Valued Logics
7.9
Truth-value Gaps: Future Contingents
Brief summary:
(7.9.1) Another motivation for holding that there are truth-value gaps are future contingents, which are statements about the future that can be uttered now but for which there presently are no facts that make them true or false, as for example: “The first pope in the twenty-second century will be Chinese” (132). (7.9.2) So some might claim that future contingents are really either true or false, and we do not know which yet. However, we will now examine Aristotle’s argument that this cannot be so. (7.9.3) Aristotle’s argument for future contingents is that they cannot be either true or false, because that would mean the futures they describe are certain and necessary, while in fact they are not. Take sentence S: “The first pope in the twenty-second century will be Chinese.” Suppose it is true. That means the first pope in the twenty-second century will in fact be Chinese. But that cannot be necessarily true, because we do not know yet. Suppose instead S is false. Then that pope will not be Chinese, necessarily, which also cannot be correct for the same reason. Either way, the future outcome would need to be necessary, but it is not, because for right now it is still contingent. Thus future contingents cannot be either true or false. (7.9.4) Objection: Aristotle’s argument, which can be illustrated as: “If S were true now, then it would necessarily be the case that the first pope in the twenty-second century will be Chinese” is ambiguous between □(A ⊃ B) (“‘if it is true now that the first pope in the twenty-second century will be Chinese, then it necessarily follows that the first pope in the twenty-second century will necessarily be Chinese”) or A ⊃ □B (“if it is true now that the first pope in the twenty-second century will be Chinese, then that the first pope in the twenty-second century will be Chinese is true of necessity.”) (7.9.5) If we take the first interpretation, □(A ⊃ B), then it would be true (because in a world, if something about the future is true now, then it cannot be otherwise that it will be false in the future of that world), but we cannot from A, □(A ⊃ B) infer that □B (because A or the conditional might not hold in other worlds and thus B may not be true in all other worlds). If we take the second interpretation, (A ⊃ □B) then we can from A, (A ⊃ □B) validly infer □B (by modus ponens), but we would not feel that it is justified to say (A ⊃ □B) in the first place (because we do not want to imply that B will happen no matter what anyway, fatalistically, regardless of A.) (My parenthetical explanations are faulty here and will be revised after the elaborations in section 11a.7.)
[Future Contingents as Motivation for Truth-Value Gaps]
[Future Contingents as Gaps]
[Aristotle’s Argument for Future Contingents]
[Ambiguity in Aristotle’s Argument for Future Contingents as Gaps]
[Neither Interpretation as Satisfactory]
Summary
[Future Contingents as Motivation for Truth-Value Gaps]
[Another motivation for holding that there are truth-value gaps are future contingents, which are statements about the future that can be uttered now but for which there presently are no facts that make them true or false, as for example: “The first pope in the twenty-second century will be Chinese” (132).]
[In the previous section 7.8 we examined one motivation for holding that there are truth-value gaps, namely, non-denoting descriptions. Now we will consider another motivation: future contingents. There are certain statements about the future that can be uttered now but for which there presently are no facts that make them true or false, as for example: “The first pope in the twenty-second century will be Chinese” (132).]
The second argument for the existence of truth-value gaps concerns certain statements about the future – future contingents. The suggestion is that statements such as ‘The first pope in the twenty-second century will be Chinese’ and ‘It will rain in Brisbane some time on 6/6/2066’ are now neither true nor false. The future does not yet exist; there are therefore, presently, no facts that makes such sentences true or false.
(132)
[Future Contingents as Gaps]
[So some might claim that future contingents are really either true or false, and we do not know which yet. However, we will now examine Aristotle’s argument that this cannot be so.]
[(ditto)]
It might be replied that such sentences are either true or false; it’s just that we do not know which yet. But there is a very famous argument, due to Aristotle, to the effect that this cannot be the case. It can be put in different ways; here is a standard version of it.
(132)
[Aristotle’s Argument for Future Contingents]
[Aristotle’s argument for future contingents is that they cannot be either true or false, because that would mean the futures they describe are certain and necessary, while in fact they are not. Take sentence S: “The first pope in the twenty-second century will be Chinese.” Suppose it is true. That means the first pope in the twenty-second century will in fact be Chinese. But that cannot be necessarily true, because we do not know yet. Suppose instead S is false. Then that pope will not be Chinese, necessarily, which also cannot be correct for the same reason. Either way, the future outcome would need to be necessary, but it is not, because for right now it is still contingent. Thus future contingents cannot be either true or false.]
[(ditto)]
Let S be the sentence ‘The first pope in the twenty-second century will be Chinese.’ If S were true now, then it would necessarily be the case that the first pope in the twenty-second century will be Chinese. If S were false now, then it would necessarily be the case that the first pope in the twenty-second century will not be Chinese. Hence, if S were either true or false now, then whatever the state of affairs concerning the first pope in the twenty-second century, it will arise of necessity. But this is impossible, since what happens then is still a contingent matter. Hence, it is neither true nor false now.
(132)
[Ambiguity in Aristotle’s Argument for Future Contingents as Gaps]
[Objection: Aristotle’s argument, which can be illustrated as: “If S were true now, then it would necessarily be the case that the first pope in the twenty-second century will be Chinese” is ambiguous between □(A ⊃ B) (“‘if it is true now that the first pope in the twenty-second century will be Chinese, then it necessarily follows that the first pope in the twenty-second century will necessarily be Chinese”) or A ⊃ □B (“if it is true now that the first pope in the twenty-second century will be Chinese, then that the first pope in the twenty-second century will be Chinese is true of necessity.”)]
[Priest next discusses a problem with this argument. But I do not understand it very well, because it involves the conditional, but in the example the conditional in question is not apparent to me. Yet perhaps we can borrow from section 7.9.5 below and say that for A ⊃ B, in our example, the ‘A’ of the conditional is ‘if it is true now that the first pope in the twenty-second century will be Chinese’ and the B in the conditional is ‘then the first pope in the twenty-second century will be Chinese’ . I probably have this wrong. At any rate, Priest says that one way to counter Aristotle’s argument is to say that it rests on a fallacy of ambiguity, where “Statements of the form ‘if A then necessarily B’ are ambiguous between ‘if A, then, it necessarily follows that B’ – □(A ⊃ B) – and ‘if A, then B is true of necessity’ – A ⊃ □B.” I am not sure still about the point. I am guessing it is the following. We are saying that our conditional is something like, “‘if it is true now that the first pope in the twenty-second century will be Chinese, then the first pope in the twenty-second century will necessarily be Chinese”. Priest says it is ambiguous whether this means □(A ⊃ B) (“‘if it is true now that the first pope in the twenty-second century will be Chinese, then it necessarily follows that the first pope in the twenty-second century will necessarily be Chinese”) or A ⊃ □B (“if it is true now that the first pope in the twenty-second century will be Chinese, then that the first pope in the twenty-second century will be Chinese is true of necessity.”) Priest then says that “Moreover, neither of these entails the other (even in Kυ).” I do not understand the point yet. But it seems he continues the treatment in following paragraph, where we will continue working through the matter.]
One might say much about this argument, but a standard, and very plausible, response to it is that it hinges on a fallacy of ambiguity. Statements of the form ‘if A then necessarily B’ are ambiguous between ‘if A, then, it necessarily follows that B’ – □(A ⊃ B) – and ‘if A, then B is true of necessity’ – A ⊃ □B. Moreover, neither of these entails the other (even in Kυ).
(132)
[Neither Interpretation as Satisfactory]
[If we take the first interpretation, □(A ⊃ B), then it would be true (because in a world, if something about the future is true now, then it cannot be otherwise that it will be false in the future of that world), but we cannot from A, □(A ⊃ B) infer that □B (because A or the conditional might not hold in other worlds and thus B may not be true in all other worlds). If we take the second interpretation, (A ⊃ □B) then we can from A, (A ⊃ □B) validly infer □B (by modus ponens), but we would not feel that it is justified to say (A ⊃ □B) in the first place (because we do not want to imply that B will happen no matter what anyway, fatalistically, regardless of A.) (My parenthetical explanations are faulty here and will be revised after the elaborations in section 11a.7.)]
[So we have ‘If S were true now, then it would necessarily be the case that the first pope in the twenty-second century will be Chinese’. Let us consider the two interpretations from section 7.9.4. {1} □(A ⊃ B). This would be true but invalid. (Let us try to assess why, but I am not certain at this point. The argument would be:
A, □(A ⊃ B) ⊨ □B
I have to guess wildly here. □(A ⊃ B) is true intuitively because we are saying what is true now about the future cannot later turn out to be false. However, we cannot from A, □(A ⊃ B) validly conclude that that □B. Let us use the tableau set-up procedure from section 2.4.2 and the tableau rules from section 2.4.4. Note that the branch closes “iff for some formula, A, and number, i, A, i and ¬A, i both occur on the branch. (It must be the same i in both cases.)” (p.25, section 2.4.5)
| A, □(A ⊃ B) ⊬ □B | ||
| 1. . 2. . 3. . 4. . 5. . 6. . 7. . 8. . . |
A,0 ↓ □(A ⊃ B),0 ↓ ¬□B,0 ↓ ◊¬B,0 ↓ 0r1 ↓ ¬B,1 ↓ (A ⊃ B),1 ↙ ↘ ¬A,1 B,1 × | P . P . P . 3¬□ . 4◊ . 4◊ . 6◊rD
7⊃ (8×1) open invalid |
(this is not in the text and is likely mistaken)
For counter-models:
Counter-models can be read off from an open branch of a tableau in a natural way. For each number, i, that occurs on the branch, there is a world, wi; wiRwj iff irj occurs on the branch; for every propositional parameter, p, if p, i occurs on the branch, vwi(p) = 1, if ¬p, i occurs on the branch, vwi(p) = 0 (and if neither, vwi(p) can be anything one wishes).
(p.27, section 2.4.7)
So we have worlds 0 and 1 where 0r1. In world 0, A is 1 (and B is 1), and in world 1, A is 0 and B is 0. So maybe the idea is that A, □(A ⊃ B) ⊨ □B is invalid, because even though in world 0, A and □(A ⊃ B) are true, and thus we can validly infer that in world 0, B is true, in other worlds A and B might not be true, and thus we cannot infer □B from A and □(A ⊃ B) in world 0. I am guessing here, sorry. {2} (A ⊃ □B). This is valid but we do not have reason to think it is true. Here I am guessing that the valid inference is:
A, (A ⊃ □B) ⊨ □B
In that case, it would seem to be valid by modus ponens. But, now we no longer have reason to think that it is true. I will guess very wildly again. Here maybe, (but likely not, sorry) the problem is sort of like saying that no matter what (and put aside the truth of A), the pope will be Chinese, almost fatalistically. I am guessing a lot. I will need to come back and revise this later, as I did not follow the thinking here, maybe after section 11a.7 when we pursue the argument further.]
Now, consider the sentence ‘If S were true now, then it would necessarily be the case that the first pope in the twenty-second century will be Chinese’, which is employed in the argument. If this is interpreted in the first way (□(A ⊃ B)), it is true, but the argument is invalid. (Since A, □(A ⊃ B) ⊭ □B.) If we interpret it in the second way (A ⊃ □B), the argument is certainly valid, but now there is no reason to believe the conditional to be true (or, if there is, this argument does not provide it). Similar considerations apply to the second part of the argument. Aristotle’s argument does not, therefore, appear to work.9
(133)
9. I will have more to say about the argument in 11a.7.
(133)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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