by Corry Shores
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[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
11a
Appendix: Many-valued Modal Logics
11a.7
Future Contingents Revisited
Brief summary:
(11a.7.1) We may use many-valued modal logics for contending with the problem of future contingents. Aristotle’s analysis of them lends to the many-valued solutions. We begin with the intuition that there are contingent future events for which there are presently no facts that could make them true or false, (as for example, “The first pope in the twenty-second century will be Chinese” (132)). We then suppose that right now a statement about a future contingent is true (or false). That would mean that it cannot be otherwise, and thus fatalism would hold if future contingents are presently either true or false. But that goes against our original intuition that nothing in the present makes such statements true or false, so Aristotle concludes that statements about future contingents cannot have either the value true or false. (11a.7.2) Priest next quotes the important Aristotle passage for this discussion of future contingents, from De Int. 18b10–16.
. . . if a thing is white now, it was true before to say that it would be white, so that of anything that has taken place, it was always true to say ‘it is’ or ‘it will be’. But if it was always true to say that a thing is or will be, it is not possible that it should not be or not come to be, and when a thing cannot not come to be, it is impossible that it should not come to be, and when it is impossible that it should not come to be, it must come to be. All then, that is about to be must of necessity take place. It results from this that nothing is uncertain or fortuitous, for if it were fortuitous it would not be necessary.
(251-252, quoting De Int. 18b10–16. Translation from Vol. 1 of Ross (1928).)
(11a.7.3) This argument may be read in the following way. “Let q be any statement about a future contingent event. Let Tq be the statement that it is (or was) true that q. Then □(Tq ⊃ q). Hence Tq ⊃ □q. And since □q is not true, neither is Tq. A similar argument can be run for ¬q. So neither Tq nor T¬q holds. Read in this way, the reasoning contains a modal fallacy (passing from □(A ⊃ B) to (A ⊃ □B))” (252). (11a.7.4) The above reading is incorrect, because Aristotle holds that the past and present are unchangeable and thus necessary. So the inference from □(Tq ⊃ q) should be □Tq ⊃ □q, which is valid. (11a.7.5) The above argument can be formulated without the conditional or the Tq formula. We have the statement about the future, q. “If q were true, this would be a present fact, and so fixed; that is, it would be necessarily true, that is: q ⊨ □q. Similarly, if it were false, it would be necessarily false: ¬q ⊨ □¬q. Since neither □q nor □¬q holds, neither q nor ¬q holds” (252). (11a.7.6) Aristotle does not allow exceptions to the Law of Non-Contradiction. So the sort of many-valued modal logic we use in application to his Future Contingents argument should validate it. Thus we should not use KFDE or KLP but rather KK3, in which there is the option for formulas to be neither true nor false, but not the option for contradictions. (11a.7.7) In our many-valued modal logic, we indicate futurity with the R accessibility relation:“Think of the accessibility statement wRw′ as meaning that w′ may be obtained from w by some number (possibly zero) of further things happening” (252). Given the nature of time, R is reflexive and transitive but not symmetrical. To capture Aristotle’s assumption that “once something is true/false, it stays so,” we will use a modified heredity constraint called the Persistence Constraint: “for every propositional parameter, p, and world, w:
If pρw1 and wRw′, pρw′1
If pρ w0 and wRw′, pρ w′0
(11a.7.8) The persistence constraint does not hold for modalized formulas. (11a.7.9) Our many-valued K3ρτ logic, augmented by the Persistence Constraint, is called A (for Aristotle). “In this logic p ⊨ □p and ¬p ⊨ □¬p. Aristotle’s argument therefore works. But, of course, in A, p ∨ ¬p may fail to be true.” (11a.7.10) For our Aristotle logic A, neither □p not □¬p holds. However, Aristotle thinks that eventually p or ¬p will have to hold, thus he thinks □(p ∨ ¬p). Yet, this does not hold in logic A. (11a.7.11) To allow □(p ∨ ¬p) to hold in logic A, we can take a temporal perspective of the end of time when everything has been decided. “Call a world complete if every propositional parameter is either true or false. A natural way of giving the truth conditions for □ is as follows:
□Aρw1 iff for all complete w′ such that wRw′, Aρw′1
□Aρw0 iff for some complete w′ such that wRw′, Aρw′ 0
The truth/falsity conditions for ◊ are the same with ‘some’ and ‘all’ interchanged. □A may naturally be seen as expressing the idea that A is inevitable. [...] for any complete world, w, Persistence holds for all formulas. It follows that at such a world, A is true iff □A is, and that all formulas are either true or false” (254). (11a.7.12) These above revised truth/falsity conditions for necessity allow us to capture the important assumptions and valid inferences in Aristotle’s argumentation regarding future contingency, namely: “p ⊨ □p, ¬p ⊨ □¬p (so Aristotle’s argument still works), ⊨ □(p ∨ ¬p), but not ⊨ □p ∨ □¬p” (254).
[Many-Valued Modal Logics and Future Contingents. Aristotle’s Analysis.]
[Aristotle’s Formulation of the Fatalism from Affirming the Present Truth of Future Contingents]
[One Possible Formal Reading of Aristotle’s Argument]
[A More Accurate Reading]
[A Simplification of the Argument]
[Using KK3]
[The R Relation and the Persistence Constraint]
[The Non-Holding of the Persistence Constraint for Modalized Formulas]
[Logic A and the Success of Aristotle’s Argument]
[A Missing Notion in Logic A]
[Using Complete Worlds to Remedy the Problem with Logic A]
[The Success of Modified Logic A]
Summary
[Many-Valued Modal Logics and Future Contingents. Aristotle’s Analysis.]
[We may use many-valued modal logics for contending with the problem of future contingents. Aristotle’s analysis of them lends to the many-valued solutions. We begin with the intuition that there are contingent future events for which there are presently no facts that could make them true or false, (as for example, “The first pope in the twenty-second century will be Chinese” (132)). We then suppose that right now a statement about a future contingent is true (or false). That would mean that it cannot be otherwise, and thus fatalism would hold if future contingents are presently either true or false. But that goes against our original intuition that nothing in the present makes such statements true or false, so Aristotle concludes that statements about future contingents cannot have either the value true or false.]
[Recall from section 7.9 our discussion of 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” (p.132, section 7.9.1). Priest notes now that many-valued modal logics can further our analysis of this philosophical problem of future contingents. Now suppose that we have a statement we utter now about a future contingent. If it is true, then in fact nothing else can happen, and if it is false, it certainly cannot happen. Aristotle argues that if statements about future contingents that are uttered now must take either the value true or the value false, then fatalism holds. For, every future event is already determined. But then, this goes against our original intuition that they really are contingent events. And so Aristotle says that statements about future contingents cannot be true or false.]
Many-valued modal logics engage with a number of philosophical controversies. Let me illustrate with respect to Aristotle’s argument concerning future contingents, which we met in 7.9. In De Interpretatione, ch. 9, Aristotle argued famously that if contingent statements about the future were now either true or false, fatalism would follow. He therefore denied that contingent statements about the future are true or false.
(252)
[Aristotle’s Formulation of the Fatalism from Affirming the Present Truth of Future Contingents]
[Priest next quotes the important Aristotle passage for this discussion of future contingents, from De Int. 18b10–16.]
[Priest next quotes the relevant passages in Aristotle’s De Int. 18b10–16. Here Aristotle explains how one might come to the fatalistic conclusion we mention above in section 11a.7.1. Someone might know how something is white right now in the present. That means, at any time in the past, the statement, “the thing will be white,” would be true. This holds for anything that happens. If it happens, then it is true to say that it is so, and it would be true in the past to say that it will be so. But someone might then infer that every future event, before it happens, were it stated, would be true. Here Aristotle introduces some notions of possibility and impossibility. So given any future event that can be stated now in advance, supposing the statement to be true, then it is impossible for that event not to happen later in the future. Thus, “All then, that is about to be must of necessity take place”. Hence nothing happens by chance.]
The argument that the law of excluded middle entails fatalism is worth quoting in detail:2
. . . if a thing is white now, it was true before to say that it would be white, so that of anything that has taken place, it was always true to say ‘it is’ or ‘it | will be’. But if it was always true to say that a thing is or will be, it is not possible that it should not be or not come to be, and when a thing cannot not come to be, it is impossible that it should not come to be, and when it is impossible that it should not come to be, it must come to be. All then, that is about to be must of necessity take place. It results from this that nothing is uncertain or fortuitous, for if it were fortuitous it would not be necessary.
(251-252)
2. De Int. 18b10–16. Translation from Vol. 1 of Ross (1928).
(251-252)
[One Possible Formal Reading of Aristotle’s Argument]
[This argument may be read in the following way. “Let q be any statement about a future contingent event. Let Tq be the statement that it is (or was) true that q. Then □(Tq ⊃ q). Hence Tq ⊃ □q. And since □q is not true, neither is Tq. A similar argument can be run for ¬q. So neither Tq nor T¬q holds. Read in this way, the reasoning contains a modal fallacy (passing from □(A ⊃ B) to (A ⊃ □B))” (252). ]
[Priest next will give a modal formulation of the above argument from section 11a.7.2. (Here he seems to be including certain sorts of tense senses, but I cannot tell if he is using a tense logic (see section 3.6a and section 3.6b .)) Let us look at the first sentence: “if a thing is white now, it was true before to say that it would be white, so that of anything that has taken place, it was always true to say ‘it is’ or ‘it will be’.” (We begin with the following structure. Something is white now. That means any statement made in the past about it being white now would be true.) So even though we are speaking about the present situation where the thing is white, we are transporting our temporal perspective into the past, and we are saying that in the past, it was true to say what is now true in the present. Now, staying in the past perspective, we have “it will be white”. Here Priest says we will call such statements about a contingent future, q. The next part read, “But if it was always true to say that a thing is or will be, it is not possible that it should not be or not come to be, and when a thing cannot not come to be, it is impossible that it should not come to be, and when it is impossible that it should not come to be, it must come to be.” Here we begin with, “if it was always true to say that a thing is or will be.” Recall that our statement of the future contingent situation is called q. Priest will have us call Tq the statement that it is now true that q (or that it was true). So in our example, our Tq would seem to be something like “it is now true that it will be white” or “it was true that it will be white.” And let us look at the second part of that sentence which was “if it was always true to say that a thing is or will be, it is not possible that it should not be or not come to be, and when a thing cannot not come to be, it is impossible that it should not come to be...”. The next formulation Priest mentions is: □(Tq ⊃ q). But I am not sure if that comes from the cited sentences or how otherwise we obtain it. Maybe we should look at the phrase, “it is not possible that it should not be” in the fuller clause, “if it was always true to say that a thing is or will be, it is not possible that it should not be or not come to be.” As far as I would guess, “not possible not” would be something like ¬◊¬ (see for example Nolt’s Logics, section 11.1), and thus we would have Tq ⊃ ¬◊¬q or Tq ⊃ □q. But that is the next formulation we are to draw from □(Tq ⊃ q). So I am still trying to figure out where that comes from. Maybe for it, we think of the Tq as being in the past and the q as now, thus the □(Tq ⊃ q) would mean maybe something like what is said in “. . . if a thing is white now, it was true before to say that it would be white, so that of anything that has taken place, it was always true to say ‘it is’ or ‘it will be’.” I am sorry that I cannot find it exactly with the language for the necessity operator. At any rate, we do get something straightforwardly like, Tq ⊃ □q when Aristotle writes, “All then, that is about to be must of necessity take place.” Priest then notes the rest of this argumentation, which seems to maybe come after our quotation, namely, that since as we assumed about future contingents that it cannot be □q, (for, were it necessary than it would no longer be contingent), that means it cannot be that Tq (by modus tollens), (so it cannot be that it our statement about the future is true now or in the past.) Priest says we can make a similar argument for ¬q. Thus “neither Tq nor T¬q holds.” Next Priest notes that when we construe the argument in this manner, we find that one of the inferences is invalid (that “the reasoning contains a modal fallacy”), namely, “passing from □(A ⊃ B) to (A ⊃ □B))”. As we saw in section 7.9, many commentators read the problem this way. (Let me just try to understand the situation. I will attempt to make the tableau for this inference, but I know it is likely mistaken:
| □(Tq ⊃ q) ⊢ Tq ⊃ □q | ||
| 1. . 2. . 3. . 4. . 5. . 6. . 7. . 8. . 9. .
. | □(Tq ⊃ q),0
↓ ¬(Tq ⊃ □q),0 ↓ Tq,0 ↓ ¬□q,0 ↓ ◊¬q,0 ↓ 0r1 ↓ ¬q,1 ↓ Tq ⊃ q,1 ↙ ↘ ¬Tq,1 q,1 × | P . P . 2¬⊃ . 2¬⊃ . 4¬□ . 5◊r . 5◊r . 1,6□r . 8⊃ (9b×7) open invalid |
(This is not in the text and it is likely mistaken.)
However, I was able to get it to close by structuring it like the “distribution axiom” which I came across here.
| □(Tq ⊃ q) ⊢ □Tq ⊃ □q | ||
| 1. . 2. . 3. . 4. . 5. . 6. . 7. . 8. . 9. . 10. .
. | □(Tq ⊃ q),0
↓ ¬(□Tq ⊃ □q),0 ↓ □Tq,0 ↓ ¬□q,0 ↓ ◊¬q,0 ↓ 0r1 ↓ ¬q,1 ↓ Tq ⊃ q,1 ↙ ↘ ¬Tq,1 q,1 ↓ × Tq,1 ×
| P . P . 2¬⊃ . 2¬⊃ . 4¬□ . 5◊r . 5◊r . 1,6□r . 8⊃ (9b×7) 3,6□r (10×9) closed valid |
(This is not in the text and it is likely mistaken.)
]
One way to read the passage is as follows. Let q be any statement about a future contingent event. Let Tq be the statement that it is (or was) true that q. Then □(Tq ⊃ q). Hence Tq ⊃ □q. And since □q is not true, neither is Tq. A similar argument can be run for ¬q. So neither Tq nor T¬q holds. Read in this way, the reasoning contains a modal fallacy (passing from □(A ⊃ B) to (A ⊃ □B)). Many commentators have read the passage thus (see 7.9).
(252)
[A More Accurate Reading]
[The above reading is incorrect, because Aristotle holds that the past and present are unchangeable and thus necessary. So the inference from □(Tq ⊃ q) should be □Tq ⊃ □q, which is valid.]
[(The part that we might find odd in the above formal reading of Aristotle’s argument in section 11a.7.3 is where it says: Tq ⊃ □q. It seems to leave out the fact the assumption that something true in the past or present cannot be changed, and thus it should really read □Tq ⊃ □q, which, as we saw in the second tableau above in section 11a.7.3, was validly inferable from □(Tq ⊃ q). Priest may be making a different point, so please check the quotation.)]
But this may not do Aristotle justice. It is clear that he thinks that the past and present are fixed (unchangeable, now inevitable). So if s is a statement about the past or present, s ⊃ □s. Hence, Tq ⊃ □Tq, and since □(Tq ⊃ q), so that □Tq ⊃ □q, it follows that Tq ⊃ □q. There is no fallacy here.
(252)
[A Simplification of the Argument]
[The above argument can be formulated without the conditional or the Tq formula. We have the statement about the future, q. “If q were true, this would be a present fact, and so fixed; that is, it would be necessarily true, that is: q ⊨ □q. Similarly, if it were false, it would be necessarily false: ¬q ⊨ □¬q. Since neither □q nor □¬q holds, neither q nor ¬q holds” (252).]
[Priest next explains how we can simplify the argument by removing the conditional and Tq. (I will not summarize it well, but it might be the following. We have q, which is already a statement about the future. We suppose it is true, making it a present fact. As such, it is fixed and thus necessarily true. So, q ⊨ □q. And if it were false, (or if its negation held, as it seems from the formulation), it would likewise be fixed and thus necessarily false (or its negation necessarily holding). But, given that q is a future contingent, by definition neither □q nor □¬q, and thus neither q nor ¬q holds. (I do not know the exact way to say why, but I would guess that when neither □q nor □¬q holds, that means neither q nor ¬q holds, because we are assuming that q ⊨ □q and ¬q ⊨ □¬q are valid, which means that it cannot be that the premises are true but the conclusion false; and since the conclusion is false, it cannot be that the premises are true.)]
In fact, we can simplify the argument. Neither Tq nor the conditional is playing an essential role. We may run the argument as follows. If q were true, this would be a present fact, and so fixed; that is, it would be necessarily true, that is: q ⊨ □q. Similarly, if it were false, it would be necessarily false: ¬q ⊨ □¬q. Since neither □q nor □¬q holds, neither q nor ¬q holds.
(252)
[Using KK3]
[Aristotle does not allow exceptions to the Law of Non-Contradiction. So the sort of many-valued modal logic we use in application to his Future Contingents argument should validate it. Thus we should not use KFDE or KLP but rather KK3, in which there is the option for formulas to be neither true nor false, but not the option for contradictions.]
[So recall that in KFDE neither the law of contradiction nor the law of excluded middle holds. When we apply the exhaustion constraint, (making it such that formulas must be either, true, false, or both) that validates excluded middle and gives us KLP (see section 11a.4.7). Or if we apply the exclusion constraint (making it such that formulas cannot be true, false, or neither), that validates non-contradiction and gives us KK3. Now, since we want to use a many-valued modal logic to understand Aristotle’s argument, and since {1} he does not allow the law of non-contradiction to be broken, (and {2} he seems to be thinking of a neither-value situation), we should not use KFDE or KLP but rather KK3.]
To do justice to Aristotle’s argument, we must take seriously the thought that some things might be neither true nor false. Since Aristotle does not countenance violations of the ‘Law of Non-Contradiction’, an appropriate logic is KK3 – or one of its normal extensions – not KFDE or KLP.
(252)
[The R Relation and the Persistence Constraint]
[In our many-valued modal logic, we indicate futurity with the R accessibility relation:“Think of the accessibility statement wRw′ as meaning that w′ may be obtained from w by some number (possibly zero) of further things happening” (252). Given the nature of time, R is reflexive and transitive but not symmetrical. To capture Aristotle’s assumption that “once something is true/false, it stays so,” we will use a modified heredity constraint called the Persistence Constraint: “for every propositional parameter, p, and world, w: If pρw1 and wRw′, pρw′1 ; If pρ w0 and wRw′, pρ w′0 .]
[Priest will now introduce our use of the accessibility relation R for our many-valued modal analysis of future contingents. Here it seems to intuitively mean something like the accessed world comes later in time, but it is worded more in term of events than temporal moments. So I am not sure how it differs from the R relation in the tense logic we saw (see section 3.6a.2). At any rate, this R relation is reflexive and transitive. (I suppose it would not be symmetrical given the arrow of time.) One of Aristotle’s assumptions is that “once something is true/false, it stays so” (Priest 252). Now recall from section 6.3.3 the heredity condition, which says that when a proposition is true in one world, it it is true in all other worlds that are accessible from it: “for every propositional parameter, p: for all w ∈ W, if vw(p) = 1 and wRw′, vw′ (p) = 1 ” (p.105 section 6.3.3). Priest now reformulates it for many-valued modal logic and calls it here the Persistence Constraint. (Maybe we can note that nothing is said for when things are neither-valued, and thus a valueless statement about a future contingent can later obtain the true or false value when that becomes determined.)]
Think of the accessibility statement wRw′ as meaning that w′ may be obtained from w by some number (possibly zero) of further things happening. Clearly, R is reflexive and transitive. According to Aristotle, once something is true/false, it stays so. We may capture the idea by the heredity condition: for every propositional parameter, p, and world, w:
If pρw1 and wRw′, pρw′1
If pρ w0 and wRw′, pρ w′0
| Call this the Persistence Constraint. The displayed conditions follow for all unmodalised formulas, as may be shown by an easy induction. (Details are left as an exercise.)
(252-253)
[The Non-Holding of the Persistence Constraint for Modalized Formulas]
[The persistence constraint does not hold for modalized formulas.]
[Priest next notes that the persistence constraint does not hold for modalized formulas. (I do not quite grasp the reasoning here, but I will work through it the best I can. Priest writes: “Let s be the sentence ‘It rains in St Andrews on 1/1/2100’. ◊s and ◊¬s are both true. But there is a possible world (indeed, a probable one!) in which s is true, and so □s is true, and ◊¬s is false.” My current (and likely mistaken) understanding is the following: when he writes, “so □s is true, and ◊¬s is false,” he means □s is true in this possible future world where it rains and ◊¬s is false in that world. But probably I am wrong. I will just make a diagram to show my understanding, and I will revise this later after I know better. I will use a sort of diagraming that was done in section 2.3.8. I will put aside the world reflexivity idea. Let us think of three moments. In the first moment, the future contingent s is not yet decided. I will put in brackets my own representation of that, even though it can simply be omitted probably. So at time 1 (T1), neither s nor its negation is true or false.
xxxT1
xxxxxxxxxxxxxx
xxxxxxxxxxxxxxxx
xxxw1
¬[sρ∅]
x[¬sρ∅]
In the next moment, T2, two possibilities can happen, either s or ¬s will be realized. We model that as two possible worlds accessible from the first.
x
xxxT1xxxxxxxxxxxxxxxT2
xxxxxxxxxxxxxxxxxxxxw2xxxxxxx
xxxxxxxxxxxxxxxx↗xxsρ1
xxxw1 xxxxxxxxxxxx¬sρ0
¬[sρ∅]xxxxxxxxxx↘x
x[¬sρ∅]xxxxxxxxxxxxxw3xxxxxxx
xxxxxxxxxxxxxxxxxxxxsρ0
xxxxxxxxxxxxxxxxxxx¬sρ1
x
And for clarity, let us assume that in each future alternative, that the persistence constraint holds and that future decided event stays as such.
x
xxxT1xxxxxxxxxxxxxxxT2xxxxxxxxxxxxxT3
xxxxxxxxxxxxxxxxxxxxw2xxxxxx→xxxxxxw4xxxxxxx
xxxxxxxxxxxxxxxx↗xxsρ1xxxxxx→xxxxxxsρ1
xxxw1 xxxxxxxxxxxx¬sρ0xxxxxx→xxxxxx¬sρ0
¬[sρ∅]xxxxxxxxxx↘x
x[¬sρ∅]xxxxxxxxxxxxxw3xxxxxxx→xxxxxxw5xxxxxxx
xxxxxxxxxxxxxxxxxxxxsρ0xxxxxx→xxxxxxsρ0
xxxxxxxxxxxxxxxxxxx¬sρ1xxxxxx→xxxxx¬sρ1
x
Since s is 1 in one possible world accessible to the first moment, and since it is 0 in another, we would seem to have these possibility values in world 1, time 1.
x
xxxT1xxxxxxxxxxxxxxxT2xxxxxxxxxxxxxT3
xxxxxxxxxxxxxxxxxxxxw2xxxxxx→xxxxxxw4xxxxxxx
xxxxxxxxxxxxxxxx↗xxsρ1xxxxxx→xxxxxxsρ1
xxxw1 xxxxxxxxxxxx¬sρ0xxxxxx→xxxxxx¬sρ0
¬[sρ∅]xxxxxxxxxx↘x
x[¬sρ∅]xxxxxxxxxxxxxw3xxxxxxx→xxxxxxw5xxxxxxx
xx◊sρ1xxxxxxxxxxxxxsρ0xxxxxx→xxxxxxsρ0
xx◊¬sρ1xxxxxxxxxxxx¬sρ1xxxxxx→xxxxx¬sρ1
x
So maybe that would show that “◊s and ◊¬s are both true”. The next part reads, “But there is a possible world (indeed, a probable one!) in which s is true, and so □s is true, and ◊¬s is false”. My best understanding of this so far is that this other possible world would be world 2 in the diagram. And maybe when he says, “so □s is true,” he means that in world 2 □s is true (and so we are not saying something here about world 1, I am guessing). In our diagram below, I added the additional world/moment to make it more visually apparent that once it happens to be so in world 2, it persists as true. And, since ¬s does not hold in moments after that, we have ◊¬s as false, too. So:
x
xxxT1xxxxxxxxxxxxxxxT2xxxxxxxxxxxxxT3
xxxxxxxxxxxxxxxxxxxxw2xxxxxx→xxxxxxw4xxxxxxx
xxxxxxxxxxxxxxxx↗xxsρ1x;x□sρ1xxxxxxsρ1
xxxw1 xxxxxxxxxxxx¬sρ0x;x◊¬sρ0xxxxxx¬sρ0
¬[sρ∅]xxxxxxxxxx↘xx
x[¬sρ∅]xxxxxxxxxxxxxw3xxxxxxx→xxxxxxw5xxxxxxx
xx◊sρ1xxxxxxxxxxxxxsρ0xxxxxx→xxxxxxsρ0
xx◊¬sρ1xxxxxxxxxxxx¬sρ1xxxxxx→xxxxx¬sρ1
xxxxxxxxxxxxxxxxxxx
Now, as far as I can understand, Priest’s point is that we can see here that the Persistence Constraint does not work for the modal operators, because ◊¬s holds as true in moment 1, but it holds as false in moment 2, world 2. (And we are assuming that it cannot be both in this sort of logic.) Or in other words, if the Persistence Constraint did hold, then ◊¬s should hold also in world two, but it does not.
x
xxxT1xxxxxxxxxxxxxxxT2xxxxxxxxxxxxxT3
xxxxxxxxxxxxxxxxxxxxw2xxxxxx→xxxxxxw4xxxxxxx
xxxxxxxxxxxxxxxx↗xxsρ1x;x□sρ1xxxxxxsρ1
xxxw1 xxxxxxxxxxxx¬sρ0x;x◊¬sρ0xxxxxx¬sρ0
¬[sρ∅]xxxxxxxxxx↘xx
x[¬sρ∅]xxxxxxxxxxxxxw3xxxxxxx→xxxxxxw5xxxxxxx
xx◊sρ1xxxxxxxxxxxxxsρ0xxxxxx→xxxxxxsρ0
xx◊¬sρ1xxxxxxxxxxxx¬sρ1xxxxxx→xxxxx¬sρ1
xxxxxxxxxxxxxxxxxxx
]
They do not hold for modalised formulas, however; nor would one expect them to. Let s be the sentence ‘It rains in St Andrews on 1/1/2100’. ◊s and ◊¬s are both true. But there is a possible world (indeed, a probable one!) in which s is true, and so □s is true, and ◊¬s is false.
(253)
[Logic A and the Success of Aristotle’s Argument]
[Our many-valued K3ρτ logic, augmented by the Persistence Constraint, is called A (for Aristotle). “In this logic p ⊨ □p and ¬p ⊨ □¬p. Aristotle’s argument therefore works. But, of course, in A, p ∨ ¬p may fail to be true.”]
[As we saw in section 11a.7.6 above, we are using a K3 logic, which means that you can have neither value but not both truth values. And we saw in section 11a.7.7 above that our R relation is reflexive and transitive, so that gives us K3ρτ. And also in that section 11a.7.7 above, we saw that our logic is augmented by the Persistence Constraint. Priest writes, “Call K3ρτ augmented by the Persistence Constraint, A (for Aristotle)” (253). (Note, the proper unicode symbol is not showing for me for some reason; so where I use this name for the Aristotle logic, A, the ‘A’ should look like this character, “Mathematical Bold Fraktur Capital A”.) As we saw in section 11a.7.5 above, for this logic of Aristotle, p ⊨ □p and ¬p ⊨ □¬p, because we cannot change what is true in the present, and thus it is necessary. Priest then gives a counter-model to show that p ∨ ¬p can fail when p is true in one future contingency and when ¬p is true in the other. Priest’s footnoted point is philosophically important but tricky for me to grasp. I cannot guess well enough at the moment what the issue is there, so I will come back to revise this. Priest writes: “Though one might object: the Persistence Constraint should hold only for those things that are genuinely about the present (w). (A sentence can be grammatically present but essentially about the future – such as the sentence ‘ “it will rain” is true’.) Enforcing the Persistence Constraint for those p that are covertly about the future may therefore be thought to be question-begging.” The only thing in my mind right now is that maybe it has something to do with what we said in section 11a.7.4 and section 11a.7.5 above. Priest wrote, “It is clear that he thinks that the past and present are fixed (unchangeable, now inevitable). So if s is a statement about the past or present, s ⊃ □s. Hence, Tq ⊃ □Tq, and since □(Tq ⊃ q), so that □Tq ⊃ □q, it follows that Tq ⊃ □q. There is no fallacy here” (252, section 11a.7.4) and “If q were true, this would be a present fact, and so fixed; that is, it would be necessarily true, that is: q ⊨ □q. Similarly, if it were false, it would be necessarily false: ¬q ⊨ □¬q. Since neither □q nor □¬q holds, neither q nor ¬q holds” (252, section 11a.7.5). My best guess is that Priest in this footnote is making the following point, but I am just guessing here. We vindicate Aristotle by making his inference valid, which involves making presently true claims necessary. But, some might object that we can have something like q ⊨ □q only when q is about things of the present and not about the future. And so, when q does happen to be about the future, when we infer q ⊨ □q, we are incorrectly thinking that a statement about a future contingent can have a truth value that persists and thus cannot be otherwise in the future and thus is necessary now. So maybe by inferring q ⊨ □q now, we are begging the question in the sense that we are claiming something is going to happen when in fact its going-to-happen is not established right now. At any rate, I will work on this later.]
Call K3ρτ augmented by the Persistence Constraint, A (for Aristotle). In this logic p ⊨ □p and ¬p ⊨ □¬p. Aristotle’s argument therefore works. But, of course, in A , p ∨ ¬p may fail to be true. Here is a simple counter-model (I omit the arrows of reflexivity):
x
xxxxxxxxxxxxxxxxxxxxxw1xxx+p
xxxxxxxxxxxxxxxxx↗xxxxxxxx→xxxxxx
–p ; –¬p xw0
xxxxxxxxxxxxxxxxx↘x
xxxxxxxxxxxxxxxxxxxxxw2xxx+¬p
xxxxxxxxxxxxxxxxxxxx
Aristotle is vindicated.3
(253)
3. Though one might object: the Persistence Constraint should hold only for those things that are genuinely about the present (w). (A sentence can be grammatically present but essentially about the future – such as the sentence ‘ “it will rain” is true’.) Enforcing the Persistence Constraint for those p that are covertly about the future may therefore be thought to be question-begging.
(253)
[A Missing Notion in Logic A]
[For our Aristotle logic A, neither □p not □¬p holds. However, Aristotle thinks that eventually p or ¬p will have to hold, thus he thinks □(p ∨ ¬p). Yet, this does not hold in logic A.]
[Recall from section 7.10.2 that Aristotle would not want all statements to be valueless, and he would want the law of excluded middle to hold when the future events happen, even if statements about future contingents would be valueless before that time. So while neither □p not □¬p holds for Aristotle, □(p ∨ ¬p) does. (In other words, given some statement about a future contingent, as it can be otherwise than what is stated, we cannot say that it is necessarily so nor can we say it is not. However, we can say that when that event is to happen, either it or its negation must happen, thus □(p ∨ ¬p).) Nonetheless, □(p ∨ ¬p) is not valid in A.]
Matters are a little more difficult than this, however, as we noted in 7.10.2. Later in the same chapter Aristotle says:4
A sea fight must either take place tomorrow, or not; but it is not necessary that it should take place tomorrow, neither is it necessary that it should not take place, yet it is necessary that it either should or should not take place tomorrow.
He is saying that, for the appropriate p, we have neither □p not □¬p. We still have □(p ∨ ¬p), however. As is easy to see, □(p ∨ ¬p) is not valid in A.
(253)
4. De Int. 19a30–32.
(253)
[Using Complete Worlds to Remedy the Problem with Logic A]
[To allow □(p ∨ ¬p) to hold in logic A, we can take a temporal perspective of the end of time when everything has been decided. “Call a world complete if every propositional parameter is either true or false. A natural way of giving the truth conditions for □ is as follows: □Aρw1 iff for all complete w′ such that wRw′, Aρw′1 ;
□Aρw0 iff for some complete w′ such that wRw′, Aρw′ 0 . The truth/falsity conditions for ◊ are the same with ‘some’ and ‘all’ interchanged. □A may naturally be seen as expressing the idea that A is inevitable. [...] for any complete world, w, Persistence holds for all formulas. It follows that at such a world, A is true iff □A is, and that all formulas are either true or false” (254).]
[Priest next gives a remedy for the problem mentioned above in section 11a.7.10, namely, that □(p ∨ ¬p) should hold. We first consider a temporal perspective where everything has happened, and thus all truth-values for p have been decided. We call such a world complete when all its propositional parameters are either true or false. We then say that a formula with the necessity operator is true in a world only if for all complete worlds accessible from it that formula is true. I would assume then that we would have □(p ∨ ¬p) being true in all evaluations, since every complete world would have either p or ¬p hold eventually. We can get the possibility true/false conditions by using the word “some”. So when we have □A, that means it is inevitable. And in a complete world, persistence holds and “A is true iff □A is.”]
The matter may be remedied by modifying the truth conditions for □. Though neither p nor ¬p may be true at a world, w, it is natural to suppose on the Aristotelian picture that the truth value of p will eventually be decided. We may therefore view things ‘from the end of time’, when everything undetermined has been resolved. Call a world complete if every propositional parameter is either true or false. A natural way of giving the truth conditions for □ is as follows:
□Aρw1 iff for all complete w′ such that wRw′, Aρw′1
□Aρw0 iff for some complete w′ such that wRw′, Aρw′ 0
The truth/falsity conditions for ◊ are the same with ‘some’ and ‘all’ interchanged. □A may naturally be seen as expressing the idea that A is inevitable. It is not difficult to show that, for any complete world, w, Persistence holds for all formulas. It follows that at such a world, A is true iff □A is, and that all formulas are either true or false. (Details are left as an exercise.)
(254)
[The Success of Modified Logic A]
[These above revised truth/falsity conditions for necessity allow us to capture the important assumptions and valid inferences in Aristotle’s argumentation regarding future contingency, namely: “p ⊨ □p, ¬p ⊨ □¬p (so Aristotle’s argument still works), ⊨ □(p ∨ ¬p), but not ⊨ □p ∨ □¬p” (254).]
[Priest lastly shows how these modified necessity conditions allow us to more completely model Aristotle’s thinking on future contingency. Recall from above, for example from section 11a.7.5, that Aristotle’s argument requires the inferences q ⊨ □q and ¬q ⊨ □¬q. The above necessity conditions produce this: p ⊨ □p, and ¬p ⊨ □¬p. So Aristotle’s controversial inference would be valid here. We saw above in section 11a.7.11 that ⊨ □(p ∨ ¬p) holds. And finally, ⊨ □p ∨ □¬p also holds. (I did not follow the reasoning. See the quotation below.)]
With the revised truth/falsity conditions for □, p ⊨ □p, ¬p ⊨ □¬p (so Aristotle’s argument still works), ⊨ □(p ∨ ¬p), but not ⊨ □p ∨ □¬p. For the first of these, if p is true at w then, by the Persistence Constraint, p holds at any complete world accessed by w. Hence □p is true at w. The argument for the second is similar. For the third, in any complete world accessed by w, either p or ¬p holds. Hence p ∨ ¬p holds, and □(p ∨ ¬p) is true at w. (Indeed, the same holds for an arbitrary formula, A.) For the last, consider the interpretation of 11a.7.9. We may suppose that all the parameters other than p also take a classical value at w1 and w2, and hence that these worlds are complete. Neither □p nor □¬p is true at w0.5
(254)
5. What one loses on this account is, of course, the validity of the inference from □A to A, even though the accessibility relation is reflexive. The inference is guaranteed to preserve truth only at complete worlds.
(254)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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