# What exactly are the identity rules in logic?

In first order logic, I have read that there are a couple of identity rules.

• If I have "a=b" does it mean that I can also write it as "b=a"?
• Is it true one-way or both?
• 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 "basic" mathematical logic axioms for identity are :

x = x (reflexivity)

x = y → y = x (symmetry)

x= y ∧ y = z → x = z (transitivity).

Thus, from a = b we can derive b = a by simmetry, and from a=b and a=c, we derive c=a from the second one by symmetry and then, from c=a and a=b, we derive c=b by transitivity, followed by b=c by symmetry again.

I know that Mauro's answer is what you were looking for. However, in group theory, the term 'identity' has another meaning, which may be the one people coming here from search engines will be looking for. For example, in an additive group (that is, a set of numbers with the + operation; e.g. (N, +)), the identity is 0, because x + 0 = x, for every x.

In logic, we can define the set B={T,F} and the group (B, ∧). That would give the identity element T, because both T∧T≡T and F∧T≡F. In the group (B, ∨) we have identity element F, for similar reasons. Also (B, ↔) has an identity; T.

Since → is not commutative, it can have both a left and a right identity. It has a left identity: T, because both T→T≡T and T→F≡F. However, it does not have a right identity because neither x=T nor x=F gives F→x≡F. Since (B, →) doesn't have a common identity, it isn't a group.

• Logic is actually a ring. Either group is not very useful. But the two groups fit together properly and each has an identity. – user9166 Nov 27 '18 at 5:10

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:

1. Rule of identity introduction. This says one can write a line such as "a=a" by invoking identity introduction.
2. 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.

References

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

Fitch, F. B. (1952). Symbolic logic.