Restricted Identity I: Examples
If you follow the Rhine upstream, you'll reach Reichenau in Switzerland, where its two tributaries, the Vorderrhein and the Hinterrhein, meet. As far as I know, it is undefined which of them, if any, is the Rhine. Obviously that's not a mystery but just a matter of stipulation. So let's stipulate that 'R1' is to denote the continuation of the Rhine through the Vorderrhein, and 'R2' its continuation through the Hinterrhein.
Now how many rivers flow through Cologne (that's much further down the Rhine)? Definitely just one. Yet both R1 and R2 flow through Cologne. How is that possible? The answer is that
At Cologne, R1 = R2.Let's call this a case of local identity.
The Amu Darya in Western Asia runs to the Aral sea. But in 1200, it ran to the Caspian, and in between it had two branches, one leading to the Aral sea and one to the Caspian. As far as I know, it is undefined which of these branches, if any, was the Amu Darya in, say, 1500. Again, this is just a matter of stipulation. Let's stipulate that 'AD1' denotes some historical continuation of the Amu Darya that ran to the Aral sea in 1500, and 'AD2' a historical continuation that ran to the Caspian in 1500.
Now how many rivers lie on the border between Turkmenistan and Uzbekistan? Just one. Yet both AD1 and AD2 do. How is this possible? Apparently,
At present, AD1 = AD2.Let's call this a case of temporal identity.
The Amazon is about 6300 km long. But that's not an essential property of the Amazon. It could have been only 5000 km long. Could it have been only 1 km long? I don't know, and I don't think careful investigation into the Amazon will reveal an answer. It's a matter of stipulation. So let's stipulate that 'A1' denotes a modal continuation of the Amazon that includes Amazon-counterparts shorter than 1 km, and 'A2' a modal continuation that doesn't include counterfactual rivers that short.
Now how many rivers with length 6300 km are there in South America? Just one. Yet both A1 and A2 are in South America and 6300 km long. How is this possible? Because
At our world, A1 = A2.Let's call this a case of contingent identity.