Here are the questions:
In first order logic, I have read that there are a couple of identity rules.
There are at least two rules for identity:
- Rule of identity introduction. This says one can write a line such as "a=a" by invoking identity introduction.
- Rule of identify elimination. Given an identity, such as "a=b" and another line containing "b" one can substitute "b" in that other line for "a" in one or more places. (See Fitch, page 81)
If I have "a=b" does it mean that I can also write it as "b=a"?
Yes, below is a proof using the two rules above, identity introduction (=I) and identity elimination (=E):
The second line can be added as a row without referencing any other line. The third referenced two lines: (1) the identity, "a=b", in line 1 and (2) the other line, "b=b", (which does not have to be an identity, but is in this case) in line 2. The second occurrence of "b" was substituted with "a" to get line 3.
Is it true one-way or both?
It is true both ways. One could extend the proof to show it is a biconditional using the same technique as above.
And if I have two statements such as "a=b" and "a=c". Can I derive both "b=c" or "c=b" from this?
The following shows that this is derivable. Line 3 is "c=b" derived by identity elimination with the identity in line 2 substituting "c" for "a" in line 1. Line 4 is "b=c" derived by identity elimination with the identity in line 1 substituting "a" for "b" in line 2.
Fitch, F. B. (1952). Symbolic logic.