Let us suppose that the following identity statement is true
(1) Byron = Arnold
(2) Byron is the originator of Frisbianity
It follows that
(3) Arnold is the originator of Frisbianity
Byron studied in New Haven
it follows that
Arnold studied in New Haven
It also follows that
If Byron enjoys a good painting, then Arnold enjoys a good painting
If Byron’s German needs work, then Arnold’s German needs work
And so on. Whatever is truthfully predicated of Byron is also true of Arnold.
Of course the converse is also true. Whatever is truthfully predicated of Arnold is also true of Byron. Again, assuming the truth of (1).
Arnold was a member of Diatheke Church
Arnold taught at NCHS
Byron was a member of Diatheke Church
Byron taught at NCHS
(4) Arnold ran off with his pastor’s wife
(5) Arnold was excommunicated from the OPC
We may conclude, assuming (1),
(6) Byron ran off with his pastor’s wife
(7) Byron was excommunicated from the OPC
There are problems with applying Leibniz’s law in certain contexts. Suppose the following is true.
Sean believes (2)
We may not conclude from this:
Sean believes (3)
For though we have stipulated the truth of (1), it may not be the case that
(8) Sean believes (1)
The lesson from this is that while Leibniz’s law applies in the predicate calculus, it does not always apply in contexts where propositions are embedded within beliefs.
Sean believes (4)
we cannot infer
Sean believes (6)
unless (8) is true.
Of course if (8) and
Sean believes (5)
were true, so would
Sean believes (7)
For further discussion about propositional attitudes see here.