Firstly we can say that logical identity is reflexive. In other words, everything (or every thing) is identical to itself. In symbols:
∀x (x = x)
For every x (or for every thing), x must equal x (or everything must be identical to itself). That is, everything has the (quasi)-relation of self-identity..
The logic of identity is also symmetrical. In symbols:
If a = b, then b = a
This can also be expressed in quantificational logic:
∀x ∀y ((x = y) ⊃ (y = x))
For every x, and for every y, if x equals — or is identical to — y, then it must also be the case that y is identical to x.
Logical identity is transitive:
If a = b, & b = c, then a = c
The above means that something is “passed on” all the way from a to c. That’s why it’s a transitive relation (or expression). Thus if a is identical to b, and b is identical to c, then a must also be identical to c - by what’s called transitivity. After all, all the symbols above simply refer to one and the same object. This too can be expressed by in quantificational terms:
∀x ∀y ∀z ((x = y) ∧ (y = z) ⊃ (x = z))
For every x, for every y, and for every z, if x is identical to y, and y is identical to z, then x must also be identical to z.
All these types of identity are satisfied by every equivalence relation (such as “is the same colour as”). Until now we’ve only talked about relations in the abstract, not specific and concrete kinds of relation. In addition, we haven’t assigned values to the symbols x, y, a and b. So with a = b we can say that the David Jones (i.e., a) and the The Thin White Duke (i.e., b) are in the relation of standing for David Bowie.
Or with a = b, and b = c, then a = c, we can say:
If Mary is the same height as John. And John is the same height as Tony. Then Mary is also 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) both b and c (or two different people).
Finally, the logic of identity satisfies Leibniz’s law (or the identity of indiscernibles). This law can be expressed in different ways, such as:
If a is identical to b, then everything true of a is also 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 which were perhaps initially taken to be two different objects. This, then, would be a de re (as opposed to de dicto) take on logical identity.
We can also say that Leibniz’s law entails the symmetrical notion (mentioned earlier) of logical identity, as expressed thus:
If a = b, then b = a
((a = b) ⊃ (b = a))
All this can be expressed quantificationally:
∀x ∀y (F) ((x = y) ⊃ F ((x) ≡ F (y)))
For every x. For every y. And also for every property (i.e., F). If x equals y, then it is the case that property F belongs to both x and y. (In this case, that property is true of.)
The identity of thing x and thing y is established by the truths applicable to x and the truths applicable to y. We can see if the two sets of truths correspond with each other. Therefore we’ve established that object x is in fact identical to object y.
The symbol ≡ can be seen in the quantificational expression above. This is the symbol for the biconditional.
The biconditional symbol is defined truth-functionally (just as with the constants and connectives). The symbol truth-functionally operates on (or is applied to) the symbols (or expressions) which surround it (or to which it applies). That is, the symbol ≡ operates on, for example, F(x) in the example above; as well as on the F(y) which follows it. More precisely, it’s truth-functional because both F(x) and F(y) both have the truth-value of either true or false. Therefore the symbol ≡ operates on their truth or falsehood in order to come out with something with a different (or the same) truth-value.
In addition, we can say that biconditional symbol ≡ is equivalent to:
((p ⊃ q) ∧ (q ⊃ p))
In other words, the biconditional is symmetrical in nature, unlike the simple logical conditional. That is, if we accept that p entails q, then we must also accept that q entails p. This is unlike this simple conditional:
(p ⊃ q) or (q ⊃ p)
This means: If p then q or if q then p. However, the symbols p ⊃ q alone don’t entail their converse: q ⊃ p. The conditional is therefore asymmetrical.
We can also say that in
p ≡ q or A ≡ B
the q (i.e., a proposition) and B (a state of affairs or an object) above offer us both the necessary and sufficient conditions for p’s and A’s truth.
Now to cite a philosophical example:
m ≡ b
This is a logical expression of mental supervenience in that for every mental change (m) there must be a physical change in the brain (b). However, clearly m and b aren’t identical otherwise we’d have m = b, not m ≡ b. This conditional is also bi because it works in two directions. Alternatively put, when anything happens in the case of m, something must also happen when it comes to b, even though m and b aren’t identical.
The conditional (i.e., not the biconditional) is a different case. That is, in m ⊃ b it’s not the case that every mental change must entail a parallel (though not identical) change in the brain. Thus:
(m ⊃ b) ¬(m ≡ b)
In terms of propositions alone (i.e., not of mental and physical changes), another way of explaining the symbol ≡ is by stating the following:
p ≡ q is true if and only if both p and q are true; otherwise it is false.
In the above, p and q stand for propositions which can be taken as being either true or false. If p is true, then q is also true; even though they aren’t identical propositions. (Similarly, if p is false.) However, if p is true and q is false (or vice versa), then the overall biconditional itself (i.e., p ≡ q) would also be false. Its falsehood would be entailed by the asymmetrical truth-values of both p and q.