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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 II:
Quantification and Identity
21
Many-valued Logics
21.9
Non-classical Identity
Brief summary:
(21.9.1) We now wonder, is it “plausible to suppose that identity statements may take non-classical values, that is, values other than 0 and 1”? (21.9.2) We might think that identity statements involving the following circumstances could take non-classical values: non-denoting terms, future contingents, verificationism, vague predicates, and paradoxes of self-reference. (21.9.3) Priest will focus here on circumstances involving vagueness with regard to identity statements. “Suppose that I have two motorbikes, a and b. Suppose that I dismantle a and, over a period of time, replace each part of b with the corresponding part of a. At the start, the machine is b; at the end, it is a. Let us call the object somewhere in the middle of the transition c. Is it true that c = a (or c = b)? It is not clear; we would seem to be in a borderline situation, so the identity predicate can be a vague one. And if one takes vague predicates to have a non-classical value (both true and false or neither true nor false) when applied to borderline cases, then there are identity statements that take such values” (468). (21.9.4) Garth Evans argues against the possibility of the borderline or vague identity circumstanced being assigned non-classical values. We first say that we will call an identity statement “indeterminate” when its truth-value is i. We next suppose that we have such an indeterminate identity statement a = b. But, since it is determinately true – it is 1 rather than i – that a = a, we can infer that a and b have different properties (for, we cannot say that a = b, because this is indeterminately true and is not 1. And, on account of the indiscernibility of identicals, because a = a, that means a has certain properties which allow it to identify with itself by means of indiscernibility. So since it has properties but since they cannot be the same as b,) we may infer that a ≠ b. “Thus, the identity is not indeterminate: it is false. There are therefore no indeterminate identities” (468). (21.9.4) Garth Evans argues against the possibility of the borderline or vague identity circumstanced being assigned non-classical values. (We first say that we will call an identity statement “indeterminate” when its truth-value is i. We next suppose that we have such an indeterminate identity statement a = b. But, since it is determinately true that a = a – it is 1 rather than i – we can infer that a and b have different properties; for, we cannot say that a = b, because this is indeterminately true and is not 1. And, on account of the indiscernibility of identicals, because a = a, that means a has certain properties which allow it to identify with itself by means of indiscernibility. So since it has properties but since they cannot be the same as b, we may infer that a ≠ b. Let me quote Priest so to have it exactly right:) “Let us say that an identity is indeterminate if the statement expressing it takes the value i. The argument goes as follows. Suppose that it is indeterminate whether a = b. It is determinately true that a = a, so a and b have different properties, and thus, a ≠ b. Thus, the identity is not indeterminate: it is false. There are therefore no indeterminate identities” (468). (21.9.5) The inference of this argument against non-classical identity is based on a contraposed form of the substitutivity of identicals. (The best I have in my own words right now, to be revised later, is: We assume that you can substitute determinately identical terms one for the other in predications, and if by making such a substitution you generate a contradiction, then the terms are not determinately identical (although they can still be indeterminately identical). We next affirm two things that we know to be true, namely, that a is indeterminately identical to b, and that a is not indeterminately equal to a. Here, on account of the substitutivity of identicals, we need to conclude that a is not determinately identical to b. For, were it so that they were determinately identical, then we would have the following contradiction, namely, that both ‘a is indeterminately equal to b’ and that ‘a is not indeterminately equal to b.’ Thus given this contradiction that ((determinately)) ‘a = b’ would cause on account of the substitutivity of identicals, we must conclude instead that a ≠ b.) (Now the correct account, all in Priest’s words:) “To analyse this argument, let us suppose that we are using one of our 3-valued logics; let us write ∇ for ‘it is indeterminate that’, and suppose that: v(∇A) ∈ D if v(A) = i ; v(∇A) = 0 otherwise . Then the argument is simply:
Suppose that ∇a = b (1)
Then since ¬∇a = a (2)
It follows that a ≠ b (3)
The inference is a contraposed form of SI; SI itself we know to be valid” (468). (21.9.6) Evans’ argument against indeterminate identity must have something wrong about how it proceeds, because the machinery of 3-valued logics do indeed allow for identity statements to take the value i. (21.9.7) Evans’ argument does not hold for gap 3-valued logics, because when a and b are distinct objects, the premises are true but the conclusion is not true: “Consider the K3 or Ł3 evaluation in which: v(=)(d, e) = 1 if v(d) = v(e) ; v(=)(d, e) = i if v(d) ≠ v(e) . Let a and b denote distinct objects. Then a = b has the value i, so ∇a = b has the value 1. a = a has the value 1, so ¬∇a = a has the value 1. But a = b and so its negation, has the value i” (468). (21.9.8) The inference under the above semantic interpretation for identity is still valid in glut logics, but under an alternate interpretation (namely, that identity statements about things that are the same are both true and false), the argument remains valid, yet it concludes that identity statements take the value i, and thus non-classical identity can hold in glut 3-valued logics: “In LP and RM3, the inference is valid, even without the second premise. Suppose that the value of ∇a = b is designated. Then the value of a = b is i. So the value of the conclusion, a ≠ b, is also designated. But this does not rule out indeterminate identity statements. Consider an LP or RM3 interpretation in which: v(=)(d, e) = i if v(d) = v(e) ; v(=)(d, e) = 0 if v(d) ≠ v(e) . Let a and b denote the same object, then (1), (2) and (3) are all designated. Yet a = b has the value i” (469).
[Wondering About Assigning Identity Statements Non-Classical Values]
[Non-Classical Values for Identity Statements in Those Cases Where It Was Fitting for Existence Statements]
[Identity and Vague Predicates: The Motorcycle Recomposition Example]
[Evans’ Argument Against Indeterminate Identity]
[A Formal Analysis of the Argument]
[Evans’ Argument Against Indeterminate Identity as Having Something Wrong to It]
[Evans’ Argument Against Indeterminate Identity Fails for Gap Logics]
[Non-Classical Identity in Glut 3-Valued Logics Where Identity Statements Can Be Both True and False]
Summary
[Wondering About Assigning Identity Statements Non-Classical Values]
[We now wonder, is it “plausible to suppose that identity statements may take non-classical values, that is, values other than 0 and 1”?]
[In the previous section 21.8 we examined the semantics for free logics that allow for identity to take values other than 0 and 1. We now wonder if this is a plausible notion.]
This raises the question of whether it is plausible to suppose that identity statements may take non-classical values, that is, values other than 0 and 1.
(468)
[Non-Classical Values for Identity Statements in Those Cases Where It Was Fitting for Existence Statements]
[We might think that identity statements involving the following circumstances could take non-classical values: non-denoting terms, future contingents, verificationism, vague predicates, and paradoxes of self-reference.]
[(ditto). (See section 21.6).]
The considerations of 21.6 about existence statements and nonclassical truth values seem to apply just as much to identity statements. I leave the reader to think about plausible candidates for non-classical identity statements in the sorts of situation discussed there.
(468)
[Identity and Vague Predicates: The Motorcycle Recomposition Example]
[Priest will focus here on circumstances involving vagueness with regard to identity statements. “Suppose that I have two motorbikes, a and b. Suppose that I dismantle a and, over a period of time, replace each part of b with the corresponding part of a. At the start, the machine is b; at the end, it is a. Let us call the object somewhere in the middle of the transition c. Is it true that c = a (or c = b)? It is not clear; we would seem to be in a borderline situation, so the identity predicate can be a vague one. And if one takes vague predicates to have a non-classical value (both true and false or neither true nor false) when applied to borderline cases, then there are identity statements that take such values” (468).]
[(ditto). (See Priest’s Logic: A Very Short Introduction ch.10 for another treatment of the motorcycle example.)]
I will just take up one of them in more detail: vagueness. Suppose that I have two motorbikes, a and b. Suppose that I dismantle a and, over a period of time, replace each part of b with the corresponding part of a. At the start, the machine is b; at the end, it is a. Let us call the object somewhere in the middle of the transition c. Is it true that c = a (or c = b)? It is not clear; we would seem to be in a borderline situation, so the identity predicate can be a vague one. And if one takes vague predicates to have a non-classical value (both true and false or neither true nor false) when applied to borderline cases, then there are identity statements that take such values.
(468)
[Evans’ Argument Against Indeterminate Identity]
[Garth Evans argues against the possibility of the borderline or vague identity circumstanced being assigned non-classical values. (We first say that we will call an identity statement “indeterminate” when its truth-value is i. We next suppose that we have such an indterminate identity statement a = b. But, since it is determinately true that a = a – it is 1 rather than i – we can infer that a and b have different properties; for, we cannot say that a = b, because this is indeterminately true and is not 1. And, on account of the indiscernibility of identicals, because a = a, that means a has certain properties which allow it to identify with itself by means of indiscernibility. So since it has properties but since they cannot be the same as b, we may infer that a ≠ b. Let me quote Priest so to have it exactly right:) “Let us say that an identity is indeterminate if the statement expressing it takes the value i. The argument goes as follows. Suppose that it is indeterminate whether a = b. It is determinately true that a = a, so a and b have different properties, and thus, a ≠ b. Thus, the identity is not indeterminate: it is false. There are therefore no indeterminate identities” (468).]
[(ditto). (Note: my reasoning in parentheses above is highly uncertain. Please consult the quotation below.)]
There is a well-known argument (due to Gareth Evans) against this possibility, however. Let us say that an identity is indeterminate if the statement expressing it takes the value i. The argument goes as follows. Suppose that it is indeterminate whether a = b. It is determinately true that a = a, so a and b have different properties, and thus, a ≠ b. Thus, the identity is not indeterminate: it is false. There are therefore no indeterminate identities.
(468)
[A Formal Analysis of the Argument]
[The inference of this argument against non-classical identity is based on a contraposed form of the substitutivity of identicals. (The best I have in my own words right now, to be revised later, is: We assume that you can substitute determinately identical terms one for the other in predications, and if by making such a substitution you generate a contradiction, then the terms are not determinately identical (although they can still be indeterminately identical). We next affirm two things that we know to be true, namely, that a is indeterminately identical to b, and that a is not indeterminately equal to a. Here, on account of the substitutivity of identicals, we need to conclude that a is not determinately identical to b. For, were it so that they were determinately identical, then we would have the following contradiction, namely, that both ‘a is indeterminately equal to b’ and that ‘a is not indeterminately equal to b.’ Thus given this contradiction that ((determinately)) ‘a = b’ would cause on account of the substitutivity of identicals, we must conclude instead that a ≠ b.) (Now the correct account, all in Priest’s words:) “To analyse this argument, let us suppose that we are using one of our 3-valued logics; let us write ∇ for ‘it is indeterminate that’, and suppose that: v(∇A) ∈ D if v(A) = i ; v(∇A) = 0 otherwise . Then the argument is simply:
Suppose that ∇a = b (1)
Then since ¬∇a = a (2)
It follows that a ≠ b (3)
The inference is a contraposed form of SI; SI itself we know to be valid” (468).]
[Priest then analyzes the argument. I do not follow it so well, so I will stumble through it a bit. It seems that overall he will show how with Evans’ argument, even if we assume that identity statements can take the value i, then we will still infer that the statements in question will take a classical value. Priest has us write “it is indeterminate that” as ∇, and the truth conditions for this operator is:
v(∇A) ∈ D if v(A) = i
v(∇A) = 0 otherwise
In other words, if a statement’s value is i, then it is indeterminate, and if its value is 1 or 0, then it is not indeterminate. Then we have three lines in the argumentation. We begin with the following supposition:
1. ∇a = b
So we begin by assuming ∇a = b, which means that a = b is indeterminate, in other words, that its value is i. Next, we say:
2. ¬∇a = a
Here maybe we are thinking the following, but I am guessing. We know that a = a is true. That means it is not i. That furthermore means that ∇a = a is false. Then, perhaps by some truth-condition for negation, then we say ¬∇a = a is true. From these two lines we conclude:
3. a ≠ b
This part is quite hard for me to follow. Priest says that the inference here is a contraposed form of the substitutivity of identicals. We saw it I think recently in section 21.8.3:
Similarly, it is not difficult to check that a=b ⊨ b=a and a=b, b=c ⊨ a=c. More generally, a = b, Ax(a) ⊨ Ax(b); for the proof of this, see 21.11.4. Note that this fact in no way depends on identities taking only classical values. Identities may well take the value i in LP or RM3 (or b in FDE).
(467)
I am not sure what is meant by the contraposed form of the substitutivity of identicals. I cannot really guess well here, but I must try. We might think of the inference of the substitutivity as saying something like, “if you have a = b, then whatever you say of a you can say of b.” Maybe the contraposed version would be (and likely not, given how much of a guess this is): “if you cannot say everything of b that you can say of a, then a ≠ b.” But even if I am on track there, I am not exactly sure yet how the inference works. We have as our premises:
1. ∇a = b
2. ¬∇a = a
and our conclusion is
3. a ≠ b
Now, maybe the inference here is simply a matter of the fact that whenever the premises are true and the conclusion is true. But without knowing the tableau rules, I am not sure how to show that. At any rate, let us try to reason through it as formally as I can make it right now, but in fact this is not really formal at all. It seems that basically we are saying the following. We assume that you can substitute determinately identical terms one for the other in predications, and if by making such a substitution you generate a contradiction, then the terms are not determinately identical to begin with (although they can still be indeterminately identical). We next affirm two things that we know to be true, namely, that a is indeterminately identical to b, and that a is not indeterminately equal to a. Here, on account of the substitutivity of identicals, we need to conclude that a is not determinately identical to b. For, were it so that they were determinately identical, then we would have the following contradiction, namely, that both ‘a is indeterminately equal to b’ and that ‘a is not indeterminately equal to b.’ Thus given this contradiction that ((determinately)) ‘a = b’ would cause on account of the substitutivity of identicals, then we must conclude instead that a ≠ b.]
To analyse this argument, let us suppose that we are using one of our 3-valued logics; let us write ∇ for ‘it is indeterminate that’, and suppose that:
v(∇A) ∈ D if v(A) = i
v(∇A) = 0 otherwise
Then the argument is simply:
Suppose that ∇a = b (1)
Then since ¬∇a = a (2)
It follows that a ≠ b (3)
The inference is a contraposed form of SI; SI itself we know to be valid.
(468)
[Evans’ Argument Against Indeterminate Identity as Having Something Wrong to It]
[Evans’ argument against indeterminate identity must have something wrong about how it proceeds, because the machinery of 3-valued logics do indeed allow for identity statements to take the value i.]
[I may not follow Priest’s next point, but maybe it is the following. The argument we saw above in 21.9.5 tried to show that we cannot have indeterminate identity statements, because, on account of the substitutivity of identicals, whenever we say that one thing is indeterminately identical to another thing, we will also need to conclude that they are determinately non-identical (thus in fact they are not indeterminately identical to begin with but are simply determinately non-identical). Priest says now that this argument must fail, because it is possible for identity statements to take the value i in these logics. But I am not sure yet what the point is there. It seems to be that since we know they can take these values, as that is built into their machinery, the problem is not with the machinery of these systems but rather with the argument used against it.]
Now it is clear that as an argument against the possibility of indeterminate identities, the argument must fail. It is quite possible for identity statements to take the value i in all these logics. What, however, is wrong with it?
(469)
[Evans’ Argument Against Indeterminate Identity Fails for Gap Logics]
[Evans’ argument does not hold for gap 3-valued logics, because when a and b are distinct objects, the premises are true but the conclusion is not true: “Consider the K3 or Ł3 evaluation in which: v(=)(d, e) = 1 if v(d) = v(e) ; v(=)(d, e) = i if v(d) ≠ v(e) . Let a and b denote distinct objects. Then a = b has the value i, so ∇a = b has the value 1. a = a has the value 1, so ¬∇a = a has the value 1. But a = b and so its negation, has the value i” (468). ]
[Let us review the three lines of the inference from section 21.9.5:
Suppose that ∇a = b (1)
Then since ¬∇a = a (2)
It follows that a ≠ b (3)
Priest says now that in logics with truth-value gaps (where i means neither true nor false), the inference from ∇a = b and ¬∇a = a to a ≠ b is invalid. Now recall also how we were evaluating sentences with the indeterminacy operator:
v(∇A) ∈ D if v(A) = i
v(∇A) = 0 otherwise
(p.468, section 21.9.5)
Priest now gives truth-conditions for the identity predicate in K3 or Ł3 (where i means neither true nor false):
v(=)(d, e) = 1 if v(d) = v(e)
v(=)(d, e) = i if v(d) ≠ v(e)
Here I am not sure what is the thinking behind this, because under this evaluation, identity statements are never false it seems, even though non-identity expressions like v(d) ≠ v(e) (which I think are saying that the domain members are not the same or not identical) can hold. In all, it seems we need to distinguish something being identical to something else, like two members of a set being the same, and an identity statement being true, false, or i. In these rules, if the members are the same, then their identity predication is true. If the members are not the same, then the identity predication is indeterminate. Still I do not know how to make intuitive sense of that, especially when in order to say that an identity statement about two things is i requires a determinate non-identity between them. At any rate, supposing all this, we find that the inference is not valid, because there is an interpretation that makes the premises true but the conclusion false. The first premise is:
Suppose that ∇a = b (1)
We will say that a and b are distinct objects, thus a ≠ b. Now recall from section 7.3.2 that in the gap logics, the only designated value is 1. And also recall that
v(=)(d, e) = i if v(d) ≠ v(e)
So the value of a = b is i. Now recall that:
v(∇A) ∈ D if v(A) = i
So that means ∇a = b is 1. Thus the first premise is true. The second premise is:
Then since ¬∇a = a (2)
Now, a = a has the value of 1, probably because:
v(=)(d, e) = 1 if v(d) = v(e)
Now recall that:
v(∇A) ∈ D if v(A) = i
v(∇A) = 0 otherwise
(p.468, section 21.9.5)
So if a = a has the value of 1, then ∇a = a has the value of 0 (because it is not i). Then, negation would flip its value, so ¬∇a = a has the value of 1. That means now that all the premise are true. But what about the conclusion? It was:
It follows that a ≠ b (3)
Now, does our interpretation make the conclusion not-true so to show the inference to be invalid? Well, recall that a = b is i. In section 7.3.2 we saw that the negation of i is i. So if a = b is i, then a ≠ b (being its negation), is also i. Therefore, the premises are true but the conclusion is not true, and so the inference is invalid. Indeterminate identity holds in gap 3-valued logics.]
That depends. Suppose, for a start, that we are in a logic with truth value gaps. Then the inference from (1) and (2) to (3) is invalid. Consider the K3 or Ł3 evaluation in which:
v(=)(d, e) = 1 if v(d) = v(e)
v(=)(d, e) = i if v(d) ≠ v(e)
Let a and b denote distinct objects. Then a = b has the value i, so ∇a = b has the value 1. a = a has the value 1, so ¬∇a = a has the value 1. But a = b and so its negation, has the value i.
(469)
[Non-Classical Identity in Glut 3-Valued Logics Where Identity Statements Can Be Both True and False]
[The inference under the above semantic interpretation for identity is still valid in glut logics, but under an alternate interpretation (namely, that identity statements about things that are the same are both true and false), the argument remains valid, yet it concludes that identity statements take the value i, and thus non-classical identity can hold in glut 3-valued logics: “In LP and RM3, the inference is valid, even without the second premise. Suppose that the value of ∇a = b is designated. Then the value of a = b is i. So the value of the conclusion, a ≠ b, is also designated. But this does not rule out indeterminate identity statements. Consider an LP or RM3 interpretation in which: v(=)(d, e) = i if v(d) = v(e) ; v(=)(d, e) = 0 if v(d) ≠ v(e) . Let a and b denote the same object, then (1), (2) and (3) are all designated. Yet a = b has the value i” (469).]
[Priest next explains why in the glut logics LP and RM3, the inference above is still valid. And he says that it is valid even without the second premise (but I do not know what is going on with the idea of excluding that premise). So again recall the argument:
Suppose that ∇a = b (1)
Then since ¬∇a = a (2)
It follows that a ≠ b (3)
Priest has us suppose for (1) that ∇a = b takes a designated value. And recall:
v(∇A) ∈ D if v(A) = i
v(∇A) = 0 otherwise
(p.468, section 21.9.5)
So ∇a = b has a designated value, and thus by the first rule, a = b is i. So the first premise is true. That means its negation is i, and so the conclusion a ≠ b is i. Recall from section 7.4.1 that in these glut logics, the designated values are 1 and i. The premises are designated values but the conclusion is also a designated value. Next it gets intuitively hard to grasp, but Priest then proposes evaluation rules that will make the inference invalid, and we will try to think philosophically about the intuitive content of this interpretation. Priest says that the evaluation rules for equality will be:
v(=)(d, e) = i if v(d) = v(e)
v(=)(d, e) = 0 if v(d) ≠ v(e)
So if two things are identically the same, then their identity predication is i (meaning here both true and false). Now, unlike before, a and b denote the same object (I think that means, a and b are constants, and are thus like names. The v function takes as their denotation the same item in the domain (or at least maybe, two items that are established as identical somehow, but I do not know yet how all that works). Thus according to our evaluation rule, since both a and b denote the same item, that means a = b has the value i. Let us see what that does to Evans’ argument:
Suppose that ∇a = b (1)
Then since ¬∇a = a (2)
It follows that a ≠ b (3)
And recall that:
v(∇A) ∈ D if v(A) = i
v(∇A) = 0 otherwise
(p.468, section 21.9.5)
The first line, ∇a = b, is a designated value (being either 1 or i, but it seems not determined which). For the next line I am not sure. I will assume that a = a is i, because of the rule:
v(=)(d, e) = i if v(d) = v(e)
And since
v(∇A) ∈ D if v(A) = i
That makes ∇a = a a designated value. But here it gets less clear to me. Suppose we say it is true. Then its negation ¬∇a = a, is false, and thus all the premises will not be designated values. It seems we need to stipulate that ∇a = a be i and not 1, but I am not sure if I am on the right track, and if indeed I am on the right track, I am not sure how that works. At any rate, somehow or other, we will see that the second line is a designated value. So our premises are all designated values. What of the conclusion? Since a = b has the value i, then its negation, a ≠ b, also has the value i. Therefore, the premises are all designated values and the conclusion is too. The important point it seems is that this argument stays valid, but under these identity evaluation semantics, its validity only leads us to conclude that a = b has the value i and thus that non-classical identity can hold in a 3-valued glut logic. But let us now examine the philosophical intuitions and implications here. We are saying that a and b in a = b denote the same object, but a = b is both true and false. It would also seem that, because of the evaluation rule
v(=)(d, e) = i if v(d) = v(e)
that a = a is both true and false. That means a ≠ b is both true and false. How might this work? I think for example of the cases of development involving vague predicates in section 11.2, section 11.3, and sections 21.6.7 and 21.6.8. But think generally of different expressions of something that is thought be the “same” on account of it being what is undergoing variation, thus both having identity in one sense and not having it in another. After working more on Priest’s philosophy of non-classical identity and multiple-denotation, I will try to say more. But for now I note that identity statements (even self-identity statements) can be both true and false, and thus that you being identical to yourself is both true and false, under this glut many-valued logic with these identity evaluation rules. This might help us with understanding the properties of identity for things that are changing. In one sense the identity holds, in that it is a flux that is a unity by tight temporal contiguities or overlaps of the parts, but it is not unity in that the changes make the composition heterogeneous over time. More on this later.]
In LP and RM3, the inference is valid, even without the second premise. Suppose that the value of ∇a = b is designated. Then the value of a = b is i. So the value of the conclusion, a ≠ b, is also designated. But this does not rule out indeterminate identity statements. Consider an LP or RM3 interpretation in which:
v(=)(d, e) = i if v(d) = v(e)
v(=)(d, e) = 0 if v(d) ≠ v(e)
Let a and b denote the same object, then (1), (2) and (3) are all designated. Yet a = b has the value i.
(469)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
