Priest (21.6) An Introduction to Non-Classical Logic, ‘Existence and Quantification,’ summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Logic and Semantics, entry directory]

[Graham Priest, entry directory]

[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 II:

Quantification and Identity

 

21

Many-valued Logics

 

21.6

Existence and Quantification

 

 

 

 

Brief summary:

(21.6.1) We can add inner and outer quantifiers to our quantified 3-valued logics. Outer quantifiers behave as normal, but inner quantifiers are  problematic when existential statements take the value i, and so inner quantifiers are primitive. (21.6.2) We now wonder if it makes sense for the existential predicate to have  non-classical values. (21.6.3) According to a certain view, we can think of existence statements of the form ℭa taking the value i under the sense of neither true nor false. (21.6.4) One argument for truth-valueless existence statements could be that non-denoting ones are valueless. “But the claim about non-denotation is not very plausible as far as the existence predicate goes. Supposing that the name ‘Sherlock Holmes’ does not denote anything, it would seem that ‘Sherlock Holmes exists’ is false, not truth-valueless” (464). (21.6.5) Another possibility is to say that existence statements can be neither true nor false when they state the existence of something bound up with a future contingency. So, we might say, “‘The first Pope of the 25th century will exist (but does not yet)’ or ‘Hilary will exist’ – where ‘Hilary’ rigidly designates the first Pope of the 25th century – is neither true nor false. But this seems wrong. If there is such a Pope, this is true” (464). (21.6.6) There is a stronger argument for truth-valueless existence statements, namely, ones that call for verificationism. So if “one can verify neither ‘a exists’ nor its negation, for some suitable a, then this statement is neither true nor false. Thus, for example, ‘The author of the Dao De Ching in fact existed’, or ‘Laozi in fact existed’ might be of this kind” (464). (21.6.7) Another way that we can have valueless existence statements would be borderline ranges of vague predicates, as for example during the gradual process of death where during a certain period some but not all vital bodily functions have ceased and thus when there is “a grey area where it is vague as to whether or not someone exists”. (21.6.8) We can also think of borderline existence cases as involving the value i with the sense of both true and false. For, “What intuition tells us, after all, is that the statement in question seems to be as true as it is false, as false as it is true; and, as far as that goes, the symmetric positions, both and neither, would seem to be as good as each other. Hence, borderline cases of existence might deliver existence statements that are both true and false” (464). (21.6.9) There are existence statements involving paradoxical self-reference that can be considered both true and false. Priest gives the example of Berry’s paradox. “Consider all those (whole) numbers that can be specified in English by a (context-independent) description with less than, say, 100 words. There is a finite number of these, so there are many numbers that cannot be so specified. There must therefore be a least. But there cannot be such a number, since if it did exist it would be specified by the description ‘the least (whole) number that cannot be specified in English by a description with less than 100 words’. The least whole number that cannot be specified in English by a description with less than 100 words both does and does not, therefore, exist” (465).

 

 

 

 

 

Contents

 

21.6.1

[Outer and Inner Domain Quantifiers in 3-Valued Logics]

 

21.6.2

[Wondering About the Sense of the Existential Predicate Taking Non-Classical Values]

 

21.6.3

[Existence Statements as Gaps]

 

21.6.4

[Existence Gaps and Non-Denotation]

 

21.6.5

[Existence Gaps and Future Contingents]

 

21.6.6

[Existence Gaps and Verificationism]

 

21.6.7

[Dying as Involving a Vague, Valueless Existential Predication]

 

21.6.8

[Borderline Existence Statements as Both True and False]

 

21.6.9

[Paradoxes of Self-Reference Involving Existence Statements, Like Berry’s Paradox, as Both True and False]

 

 

 

 

 

Summary

 

21.6.1

[Outer and Inner Domain Quantifiers in 3-Valued Logics]

 

[We can add inner and outer quantifiers to our quantified 3-valued logics. Outer quantifiers behave as normal, but inner quantifiers are  problematic when existential statements take the value i, and so inner quantifiers are primitive.]

 

[We will now recall some matters from section 13.5  regarding the useful distinction and addition of inner and outer quantifiers:

(13.5.1) We might want a free logic where quantifiers range over all objects and not just existent ones. (13.5.2) Quantifiers ranging over the outer domain D are called the outer quantifiers, and they are written as ∃ and ∀. The quantifiers that range over the inner domain E are called inner quantifiers, and they are written as ∃^E and ∀^E. (13.5.3) We read ∀xA as ‘Every x is such that A’; ∀^ExA as ‘Every existent x is such that A’; ∃xA as ‘Some x is such that A’ or as ‘Something is A’; and ∃^ExA as ‘there exists an x such that A’ or as ‘there is an x such that A’. (13.5.4) We should not think that the existential quantifier of natural language necessarily implies existence. (13.5.5) There is an argument for reading the existential quantifier as “there exists”. The argument wants to avoid problems like the ontological argument, so it does not allow existence to be a predicate. Instead, it sees as the only other viable option for expressing existence as being the existential quantifier. Part of the thinking is that only things that are there can be predicated. But this is not a convincing argument, because there are many examples of predication of non-existing objects, like Zeus being worshipped. (13.5.6) If we wish, we can define inner quantifiers in terms of outer ones, which means that “in a free logic with outer quantifiers, we can dispense with inner quantifiers altogether,” namely, in the following way:

∃^ExA     ∃x(ℭx ∧ A)

∀^ExA     ∀x(ℭx ⊃ A) 

(p297). However, “There is no way of defining outer quantifiers in terms of inner quantifiers” (297). (13.5.7) These new semantics make one problematic inference no longer problematic, namely, A_x(a) ⊨ ∃xA, now meaning that if something can be predicated, it is either an existent or non-existent object (previously it implied that any predicable thing must be existent). But it may not make the logical truth ∃x(A ∨ ¬A) unproblematic (it implies now that there must be at least a non-existent object, while before it implied there must be at least an existent object.)

(From the brief summary of section 13.5)

Priest notes now that for our 3-valued logics, we can have outer-quantifiers ranging over the whole of the domain D. But, for certain technical reasons (see below, as I have not yet tried to grasp and summarize them yet), we cannot define the inner quantifiers by means of outer-ones in 3-valued logics where existence predicates take non-classical values (like i), and so “inner quantifiers will have to be taken as primitive.” (I am not sure yet how that works, but it seems in the next section we should figure that out as an exercise.) ]

As with the two-valued case, in the free 3-valued logics we have been talking about, one can have outer quantifiers, ranging over the whole of D. The definability of the inner (existentially loaded) quantifiers in terms of the outer quantifiers and the existence predicate is, however, more problematic. If, as in 13.5.3, we write the outer quantifiers as ∀ and ∃, and use a superscript E to indicate the existentially loaded quantifiers, what we require is:

1. v(∃^ExA) = v(∃x(ℭx ∧ A))

2. v(∀^ExA) = v(∀x(ℭx ⊃ A))

We know that v(ℭk_d) ∈ D iff d ∈ E. If ℭ is a classical predicate, in the sense that for all d ∈ D, v(ℭkd) = 1 or v(ℭkd) = 0, these equations hold. The details are straightforward, and left as an exercise. (Check that if the lefthand side is 1, so is the righthand side. Then check the opposite direction. Do the same thing for 0. The case for i then follows.) If, however, existential statements may take the value i, things may go wrong. Consider an interpretation with | two members, d and e, as follows:

 

       D

+---------------+
|               |
|     E         |
|  +-----+      |
|  |     |      |
|  |  e  |   d  |
|  |     |      |
|  +-----+      |
+---------------+

If v is as follows:

 

	v(ℭ)	v(P)
d	i	1
e	1	0

 

this is a K₃ and Ł₃ interpretation. It is not difficult to check that v(∃^ExPx) = 0, but v(ℭk_d ∧ Pk_d) = i = v(∃x(ℭx ∧ Px)).

 

If v is as follows:

 

	v(ℭ)	v(P)
d	0	0
e	i	1

 

this is an LP and RM₃ interpretation. It is not difficult to check that v(∃^ExPx) = 1, but v(ℭk_e ∧ Pk_e) = i = v(∃x(ℭx ∧ Px)). Hence, if the existence predicate is allowed to take non-classical values, inner quantifiers will have to be taken as primitive.

(463)

[contents]

 

 

 

 

 

 

21.6.2

[Wondering About the Sense of the Existential Predicate Taking Non-Classical Values]

 

[We now wonder if it makes sense for the existential predicate to have  non-classical values.]

 

[(ditto)]

Arranging for this is a simple matter, and left as an exercise. However, it does raise the question of whether it makes sense for the existence predicate to have a non-classical value, the answer to which is not so obvious.

(463)

[contents]

 

 

 

 

 

 

21.6.3

[Existence Statements as Gaps]

 

[According to a certain view, we can think of existence statements of the form ℭa taking the value i under the sense of neither true nor false.]

 

[(ditto)]

Suppose that we are in a logic where i is interpreted as neither true nor false. Could a sentence of the form ℭa take this value? The answer depends on what sorts of thing one takes to be neither true nor false; but on certain views about this, the answer could be ‘yes’.

(463)

[contents]

 

 

 

 

 

 

21.6.4

[Existence Gaps and Non-Denotation]

 

[One argument for truth-valueless existence statements could be that non-denoting ones are valueless. “But the claim about non-denotation is not very plausible as far as the existence predicate goes. Supposing that the name ‘Sherlock Holmes’ does not denote anything, it would seem that ‘Sherlock Holmes exists’ is false, not truth-valueless” (464).]

 

[(ditto)]

Some have argued that a sentence containing a non-denoting name has no truth value (see 7.8). If this is the case, and a does not denote anything, | ℭa has no truth value. But the claim about non-denotation is not very plausible as far as the existence predicate goes. Supposing that the name ‘Sherlock Holmes’ does not denote anything, it would seem that ‘Sherlock Holmes exists’ is false, not truth-valueless.

(463-464)

[contents]

 

 

 

 

 

 

21.6.5

[Existence Gaps and Future Contingents]

 

[Another possibility is to say that existence statements can be neither true nor false when they state the existence of something bound up with a future contingency. So, we might say, “‘The first Pope of the 25th century will exist (but does not yet)’ or ‘Hilary will exist’ – where ‘Hilary’ rigidly designates the first Pope of the 25th century – is neither true nor false. But this seems wrong. If there is such a Pope, this is true” (464).]

 

[(ditto)]

Aristotle argued that statements about a future state of affairs that is not, as yet, determined are neither true nor false (see 7.9). If this is correct then, arguably, ‘The first Pope of the 25th century will exist (but does not yet)’ or ‘Hilary will exist’ – where ‘Hilary’ rigidly designates the first Pope of the 25th century – is neither true nor false. But this seems wrong. If there is such a Pope, this is true.

(464)

[contents]

 

 

 

 

 

 

21.6.6

[Existence Gaps and Verificationism]

 

[There is a stronger argument for truth-valueless existence statements, namely, ones that call for verificationism. So if “one can verify neither ‘a exists’ nor its negation, for some suitable a, then this statement is neither true nor false. Thus, for example, ‘The author of the Dao De Ching in fact existed’, or ‘Laozi in fact existed’ might be of this kind” (464).]

 

[(ditto)]

Better arguments can be found if one subscribes to verificationism of some kind. This might be a philosophy of mathematics which identifies mathematical truth with provability; or it might be a philosophy of science which identifies truth with empirical verifiability. If one subscribes to such a view, and one can verify neither ‘a exists’ nor its negation, for some suitable a, then this statement is neither true nor false. Thus, for example, ‘The author of the Dao De Ching in fact existed’, or ‘Laozi in fact existed’ might be of this kind.

(464)

[contents]

 

 

 

 

 

 

21.6.7

[Dying as Involving a Vague, Valueless Existential Predication]

 

[Another way that we can have valueless existence statements would be borderline ranges of vague predicates, as for example during the gradual process of death where during a certain period some but not all vital bodily functions have ceased and thus when there is “a grey area where it is vague as to whether or not someone exists”.]

 

[(ditto)]

As another example: some have argued that statements about the borderline range of some vague predicate are neither true nor false (see 11.3.6, 11.3.7). Thus, ‘Dana is an adult’, said of Dana around puberty, might be thought to be neither true nor false. But can existence be a vague predicate? Certainly: when people die they go out of existence (let us suppose). But dying can be a gradual process. Bodily functions do not normally all cease at once; there can therefore be a grey area where it is vague as to whether or not someone exists.

(464)

[contents]

 

 

 

 

 

 

21.6.8

[Borderline Existence Statements as Both True and False]

 

[We can also think of borderline existence cases as involving the value i with the sense of both true and false. For, “What intuition tells us, after all, is that the statement in question seems to be as true as it is false, as false as it is true; and, as far as that goes, the symmetric positions, both and neither, would seem to be as good as each other. Hence, borderline cases of existence might deliver existence statements that are both true and false” (464).]

 

[(ditto) (Note: the argument of gaps for dying from section 21.6.7 above seems odd to me, and the gluts version here seems much more reasonable. If it is neither true nor false that one is alive (exists), and it is neither true nor false that one is not alive, than what can we say about the person’s state of being? Can we say that it is true it is some third state? To me it seems more reasonable to say it is both true and false that one exists when in the transitional process of dying, as there is not really another predicate I can think of that applies here that would be just true (and surely some predicate or other regarding its state of being should hold, because we cannot simply say they are dead, but the dying person is still there in some state of being that is also not life, under these gap assumptions). The gap thinking seems to be the following. Can we say the dying person is dead (does not exist)? No, because they are not dead enough to be such. Can we say they are alive? No, because they are not alive enough to be such. In other words, there is a window during which neither the predicate “exists” nor its negation holds. But, as I pointed out, there is still a person there in some state of being, and presumably that state can be given a name and serve as a predication to the dying person. Now, under this gap reasoning, that predicate cannot be “exists”. But I have two problems with that. If it is not “exists,” then you are saying it is false that they exist. That to me seems like you are saying that “they exist” is false and not valueless (otherwise you might be saying they exist only partly, but then you are using a fuzzy value or maybe even a glut, which is not what we are assuming here for gaps). My other problem is that if you insist that “they exist” is neither true nor false, but that moments later after they fully die, “they exist” is false, then, as I noted, something true can be predicated of their existential state which is between existence and non-existence. So my final point here is that to be in a state between existence and non-existence would not be like jumping into a third state that is completely different from existence and non-existence but would rather seem to have certain properties of existence and certain properties of non-existence. For, it is a continuous variation from the one state to the other. So for example, you may have consciousness but not cell life-sustenance on account of stopped blood flow, or maybe you do not have consciousness but you have blood flow. I am not sure what really is involved in death processes. At any rate, it seems to me that existence in the dying transition phase would seem to be a glut, in that it is both true that you exist (in that you have enough functions at this very moment to say that you have not completely passed out of existence and thus that you still are existing, even if barely so) but it is also true to say that you do not exist (as you have a lack of certain functions that will sustain your existence for much longer and thus you are practically dead. To put it phenomenologically, when you are passing away, you will be consciousness of your own fading consciousness, and thus in one instant you will experience both your state of existence and non-existence simultaneously).)]

What of a logic where i is interpreted as both true and false. Could a sentence of the form ℭa be both true and false? Some have suggested that the statements about the borderline range of some vague predicate are both true and false. What intuition tells us, after all, is that the statement in question seems to be as true as it is false, as false as it is true; and, as far as that goes, the symmetric positions, both and neither, would seem to be as good as each other. Hence, borderline cases of existence might deliver existence statements that are both true and false.

(464)

[contents]

 

 

 

 

 

 

 

21.6.9

[Paradoxes of Self-Reference Involving Existence Statements, Like Berry’s Paradox, as Both True and False]

 

[There are existence statements involving paradoxical self-reference that can be considered both true and false. Priest gives the example of Berry’s paradox. “Consider all those (whole) numbers that can be specified in English by a (context-independent) description with less than, say, 100 words. There is a finite number of these, so there are many numbers that cannot be so specified. There must therefore be a least. But there cannot be such a number, since if it did exist it would be specified by the description ‘the least (whole) number that cannot be specified in English by a description with less than 100 words’. The least whole number that cannot be specified in English by a description with less than 100 words both does and does not, therefore, exist” (465).]

 

[(ditto)]

One final example. Some have argued that paradoxical sentences generated by the paradoxes of self-reference are both true and false (see 7.7). Some of these can be existence statements, as in Berry’s paradox, which is as follows. Consider all those (whole) numbers that can be specified in English by a (context-independent) description with less than, say, 100 words. There is a finite number of these, so there are many numbers that cannot be so specified. There must therefore be a least. But there cannot be such a number, since if it did exist it would be specified by the description ‘the least (whole) number that cannot be specified in English by a description with less than 100 words’. The least whole number that cannot be specified in English by a description with less than 100 words both does and does not, therefore, exist. So paradoxes of self-reference may deliver existence statements that are both true and false.

(465)

[contents]

 

 

 

 

 

 

 

From:

 

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

 

 

 

 

Priest (11a.7) An Introduction to Non-Classical Logic, ‘Future Contingents Revisited,’ summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

[Logic and Semantics, entry directory]

[Graham Priest, entry directory]

[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

 

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. 18^b10–16.

. . . 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. 18^b10–16. 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 T_q be the statement that it is (or was) true that q. Then □(T_q ⊃ q). Hence T_q ⊃ □q. And since □q is not true, neither is T_q. A similar argument can be run for ¬q. So neither T_q 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 □(T_q ⊃ q) should be □T_q ⊃ □q, which is valid. (11a.7.5) The above argument can be formulated without the conditional or the T_q 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 K_FDE or K_LP but rather K_K3, 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 pρ_w1 and wRw′, pρ_w′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: 

□Aρ_w1 iff for all complete w′ such that wRw′, Aρ_w′1

□Aρ_w0 iff for some complete w′ such that wRw′, Aρ_w′ 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).

 

 

 

 

 

 

Contents

 

11a.7.1

[Many-Valued Modal Logics and Future Contingents. Aristotle’s Analysis.]

 

11a.7.2

[Aristotle’s Formulation of the Fatalism from Affirming the Present Truth of Future Contingents]

 

11a.7.3

[One Possible Formal Reading of Aristotle’s Argument]

 

11a.7.4

[A More Accurate Reading]

 

11a.7.5

[A Simplification of the Argument]

 

11a.7.6

[Using K_K3]

 

11a.7.7

[The R Relation and the Persistence Constraint]

 

11a.7.8

[The Non-Holding of the Persistence Constraint for Modalized Formulas]

 

11a.7.9

[Logic A and the Success of Aristotle’s Argument]

 

11a.7.10

[A Missing Notion in Logic A]

 

11a.7.11

[Using Complete Worlds to Remedy the Problem with Logic A]

 

11a.7.12

[The Success of Modified Logic A]

 

 

 

 

 

Summary

 

11a.7.1

[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)

[contents]

 

 

 

 

 

 

11a.7.2

[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. 18^b10–16.]

 

[Priest next quotes the relevant passages in Aristotle’s De Int. 18^b10–16. 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. 18^b10–16. Translation from Vol. 1 of Ross (1928).

(251-252)

[contents]

 

 

 

 

 

 

11a.7.3

[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 T_q be the statement that it is (or was) true that q. Then □(T_q ⊃ q). Hence T_q ⊃ □q. And since □q is not true, neither is T_q. A similar argument can be run for ¬q. So neither T_q 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 T_q the statement that it is now true that q (or that it was true). So in our example, our T_q 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: □(T_q ⊃ 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 T_q ⊃ ¬◊¬q  or T_q ⊃ □q. But that is the next formulation we are to draw from □(T_q ⊃ q). So I am still trying to figure out where that comes from. Maybe for it, we think of the T_q as being in the past and the q as now, thus the □(T_q ⊃ 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,  T_q ⊃ □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 T_q (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 T_q 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 T_q be the statement that it is (or was) true that q. Then □(T_q ⊃ q). Hence T_q ⊃ □q. And since □q is not true, neither is T_q. A similar argument can be run for ¬q. So neither T_q 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)

 

[contents]

 

 

 

 

 

 

11a.7.4

[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 □(T_q ⊃ q) should be □T_q ⊃ □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: T_q ⊃ □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 □T_q ⊃ □q, which, as we saw in the second tableau above in section 11a.7.3, was validly inferable from □(T_q ⊃ 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, T_q ⊃ □T_q, and since □(T_q ⊃ q), so that □T_q ⊃ □q, it follows that T_q ⊃ □q. There is no fallacy here.

(252)

[contents]

 

 

 

 

 

 

11a.7.5

[A Simplification of the Argument]

 

[The above argument can be formulated without the conditional or the T_q 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 T_q. (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 T_q 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)

[contents]

 

 

 

 

 

 

11a.7.6

[Using K_K3]

 

[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 K_FDE or K_LP but rather K_K3, in which there is the option for formulas to be neither true nor false, but not the option for contradictions.]

 

[So recall that in K_FDE 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 K_LP (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 K_K3. 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 K_FDE or K_LP but rather K_K3.]

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 K_K3 – or one of its normal extensions – not K_FDE or K_LP.

(252)

[contents]

 

 

 

 

 

 

11a.7.7

[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 pρ_w1 and wRw′, pρ_w′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 v_w(p) = 1 and wRw′, v_w′ (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 pρ_w1 and wRw′, pρ_w′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)

[contents]

 

 

 

 

 

 

11a.7.8

[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)

[contents]

 

 

 

 

 

 

 

11a.7.9

[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, T_q ⊃ □T_q, and since □(T_q ⊃ q), so that □T_q ⊃ □q, it follows that T_q ⊃ □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)

[contents]

 

 

 

 

 

 

11a.7.10

[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)

[contents]

 

 

 

 

 

 

11a.7.11

[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:  □Aρ_w1 iff for all complete w′ such that wRw′, Aρ_w′1 ;

□Aρ_w0 iff for some complete w′ such that wRw′, Aρ_w′ 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:

□Aρ_w1 iff for all complete w′ such that wRw′, Aρ_w′1

□Aρ_w0 iff for some complete w′ such that wRw′, Aρ_w′ 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)

[contents]

 

 

 

 

 

 

11a.7.12

[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)

[contents]

 

 

 

 

 

 

 

From:

 

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.