Second order vagueness, identity, and biological species: Part II (Vague Identity)

Gareth Evans (1978) published a well known article suggesting that to accept vague objects leads to a paradox in their identity relations. With some expansion of the later steps in the proof, where Evans moves more quickly than I feel comfortable following, the argument can be summarized as follows:

Let (vague) be symbolized as a nabla with ∇,and let this be a specifier that an object or its relations are vague. Let the Delta, Δ, symbol then indicate that a relation with that object is definite. Also, specify that ∇x ↔ ¬Δx and Δx ↔ ¬∇x. Assume that two vague objects, a and b, are possibly equal, but that it is then vague that they are equal. Then

I. ∇ a = b

The next part of Evan's argument is rather tricky, and involves running a lambda substitution backwards. A lambda substitution means we take a formula or sentence in logic and modify the formula by replacing each occurrence of a variable, like x, with another value or phrase. The method is called a lambda function, and is written like this:

λx | expression | y

where y is substituted into the expression within the straight brackets wherever x is found. So for example,

λx | 2 x + 1 | 5 = 2(5) + 1 = 10 + 1 = 11.

Now, to run the lambda substitution backwards. We can see that with the above expression,

9 = 2(4) + 1 = λx | 2 x + 1 | 4

Here we will specifically consider a lambda function M, which we define as λx | a = x |.

Now, consider the lambda function λx | a = x | b which can be written by substituting b for x as a = b. Similarly, we can say, given that ∇ a = b, that:

II. ∇ λx| a = x | b, or ∇ M b

Next, Evans considers that by identity,

III. Δa = a

which he rewrites, again via a reverse lambda transformation, as

IV. Δ λx| a = x | a, or Δ M a

Now, if the vague, ∇ and the definite, Δ are considered logical inverses, we then can transform IV into

V. ¬ ∇ M a

Combining V. with I., we get

VI. ¬ ∇ M a AND ∇ M b

So, by the non-identity of non-indiscernibles, we have

VII. a ≠ b.

...which produces logical contradiction unless we allow ∇ and Δ to not be true inverses of each other.

Note here that the final conclusion, a ≠ b, drops the vague and definite modal qualifiers. This is because it is a meta-statement about the identity relation's truth values, independently of the object's vagueness. Fortunately, Evan's contradiction is avoided if we allow the the definite to be a special case, or subset, of the vague. In that case, the inverse of vague identity is not crisp or definite identity, but just nil, the null set of lack of all identity. We can define inequality for sets such that two rough sets with the same upper and lower bounds are not generally exactly equal, but may be vaguely equal, so that both I and VII can be true of two rough sets.

Any given rough set a is always such that a = a, but even if rough set a and rough set b have the same bounds, they may not contain the same elements within their vague regions (though they might), and thus it is also true in general that rough set a is nonequal to b. Of course, a may be strictly equal to b if they are both empty sets or if they have the same elements within their lower bounds and no elements within the vague zone between their upper and lower bounds. Such sets would of course be crisp, or ordinary, sets as well as vague sets, since we are allowing conventional sets as a special case of vague sets.