Sunday, 18 May 2014

The Basics of Logical Identity

Identity can be taken as a relation when taken grammatically, not logically. In other words, if two things are identical, then that identity is a relation – a relation between, say, a and b. And if we talk of self-identity, then there is a quasi-relation between a and a, or a and b if two names refer to numerically the same thing. That identity then is a relation between a given thing and itself, hence ‘quasi’. However, if we talk in terms of ‘a’ and ‘b’ being names for the same thing, then it makes more sense to talk of the relation of self-identity because it involves not only a relation between an object and itself, but one also between two different names. We can call it, then, a de dicto relation, not the de re relation of an object’s self-identity. However, despite these games with self-identity, we can still justifiably say that the relation of self-identity is not, in fact, a genuine relation. Despite that it can be taken as a genuine relation, but only of looked at purely logically. How is that?

First, we can say that identity is symmetrical. In symbols:

If a = b, then b = a

This is too self-evident to deserve commentary. However, it can be expressed in quantificational logic:

(x) (y) ((x = y) ⊃. (y = x))

To translate. For every thing x, and for every thing y, if x equals - or is identical - to thing y, then it must also be the case that thing y is identical to thing x.

We can also say that identity is reflexive. That is, everything is identical to itself (i.e., self-identity). In symbols:

(x) (x = x)

To translate: For every thing x, or for everything, x must equal thing x, or everything must equal itself. That is, everything has the quasi-relation of self-identity.

Identity is transitive:

If a = b, and b = c, then a = c

This means, firstly, that something is ‘past one’ all the way from a to c. That’s why it is a transitive relation or expression. If a is identical to b, and b is identical to c, then a too must be both identical to b and identical to c, by transitivity. After all, all the symbols simply refer to one and the same object. This too can be expressed by a quantificational expression.


(x) (y) (z) ((x = y) ∧ (y = z) ⊃. (x = z))

To translate: For everything x, for every thing y, and for every thing z, if x is identical to y, and y is identical to z, then x must also be identical to z.

All these conditions or definitions of kinds of identity are satisfied by every ‘equivalence relation’, such as ‘is the same colour as’. Up till now, in other words, we have only talked about relations in the abstract, not specific and concrete kinds of relation. In addition, we have not assigned values to the letter-symbols ‘x’, ‘y’ and so on. So with x = y we can say that the names ‘Tony Blair’ and ‘Tony Boy’ stand in the relation of standing for Tony Blair. Or, with a = b, and b = c, then a = c, we can say that if Mary is the same height as John. And John is the same height as Tony. Then Mary too is the same height as Tony, as well as being the same height as John. Mary, or a, has the relation of being the same height as, in this case, as both b and c, or two different people.

Finally, identity satisfies Leibniz’s law. What is that law? Although it can be expressed in different ways, we can express it thus:

If a is the same as b, then everything true of a is true of b.

In this version, Leibniz’s law is expressed with reference to the semantic property of truth. We can, instead, express it in terms only of the properties of identical objects, taken initially, perhaps, to be different objects. This, then, would be an ontological expression of the law, or a de re version of it. We can also say that Leibniz’s law also entails the symmetrical notion of the identity-relation expressed thus:

If a = b, then b = a

That is

((a = b) ⊃ (b = a))

then everything true of a above, and of b, is also true of b, or of what the symbol ‘b’ stands for.

So the above begins with a de re identity-relation and ends with a natural language expression or a semantic identity-relation (i.e., one that uses the concept of truth). All this can be expressed quantificationally thus:

(x) (y) (F) ((x = y) ⊃. F ((x) ≡ F (y)))

To translate: For every thing x. For every thing y. And also for every property of falsehood. If x equals y. Then it is false that every thing false of x it is also the case if and only if it is also the case that such falsehoods are true of y.

This is an example of a negative version of Leibniz’s law in that it uses the notion of ‘false of’ rather than ‘true of’. The identity of thing x and thing y is established by the falsehoods applicable to x and the falsehoods applicable to y. And then by seeing if the two sets of falsehoods correspond with one another. If they do, then we have established that object x is in fact identical to object y.

In addition, the symbol ‘≡’ is defined truth-functionally, just as is the case with the constants or connectives. It truth-functionally operates on, or is applied to, the symbols and expressions which surround it or to which it applies. That is, ‘≡’ operates on, for example, F(x) in the example above, as well as the F(y) that follows it. It is truth-functional, more precisely, because both F(x) and F(y) both have the truth-value of either true or false. Therefore ‘≡’ operates on their truth or falsehood to come out with something with a different, or the same, truth-value.

In addition, we can say that ‘≡’ is equivalent to:

((pq) ∧ (q p))

In other words, the biconditional is symmetrical in nature, unlike the simple conditional. That is, if we accept that p entails q, then we must also accept that q entails p. Or:

((pq) ∨ (qp))

This is unlike the simple conditional:

(pq) or (qp)

This means if p entails q, or if q entails p. But p q does not entail its converse:

q p. The conditional is not symmetrical but asymmetrical. We can also say that in

pq or AB

the ‘q’ and the ‘B’ offer us both the necessary and sufficient condition for p’s and A’s truth, or for their reality in terms of condition A. So p q can be a logical expression of mental supervenience in that for every mental change (p) there must be a physical change (q). Or for every mental change p is must be the case if and only if there is also a parallel physical change in the brain. However, clearly p and q are not identical otherwise we would have p ≡ q not p q. The conditional is ‘bi’ because it works in two directions or ways, or that everything that happens to p must also happen to q, even though p and q are not identical. That is, it is conditional on the nature or truth of the binary terms ‘p’ and ‘q’. The condition can be called ‘monadic’ not ‘binary’. That is, in p q it is not the case that every that happens to p must entail parallel, but not identical to, happenings with q. Instead p and q are individuals or they do not have a symmetrical relation to one another. That is:

((pq) ¬ ≡ (qp))

Another way of explaining ‘≡’ is by saying:

p q is true if and only if p and q are both true or both false; otherwise it is false.

In the above, ‘p’ and ‘q’ stand for propositions that can be taken as either true or false. If everything of p is true, then everything of p is also true, even though they are not identical. Similarly, if everything of p is false. However, if everything said of p is true and everything said of q is false, or vice versa, in the above, then the overall conditional, p q, itself would also be false. Its falsehood would be entailed by the asymmetrical truth-values of p and q.

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