No logics ever really "define" truth, they use it. It is assumed that there is some pre-theoretic understanding of what "truth" is.

But you don't even need a notion of truth. You can get by with any designated values. In mathematical logic the truth values are typically "1" and "0". Now, these are generally taken to code truth and falsity but that is not required. All that is required is that you have a designated value so that you can definition a notion of a valid inference as once that preserves designated values.

In many valued logics they will often have more than one designated value. Also, it is hard to see what the values in fuzzy logic would be. Are they "degrees" of truth? Does truth come in degrees?

A quote from Russell's Principles of Mathematics seems appropriate here:

In addition to these [indefinable primitives of mathematics], mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth.

I think that much the same can be said of logic, especially math logic. The study of truth is the domain of truth theory. See the SEP article on Truth.

I really can't state with confidence that informal logic is the same, using rather than defining truth. But a quick scan of the SEP article on Informal Logic makes me think that what I've said probably holds of informal logic as well.

No logics ever really "define" truth, they use it. It is assumed that there is some pre-theoretic understanding of what "truth" is.

But you don't even need a notion of truth. You can get by with any designated values. In mathematical logic the truth values are typically "1" and "0". Now, these are generally taken to code truth and falsity but that is not required. All that is required is that you have a designated value so that you can define a notion of a valid inference as one that preserves designated values.

In many valued logics they will often have more than one designated value. Also, it is hard to see what the values in fuzzy logic would be. Are they "degrees" of truth? Does truth come in degrees?

A quote from Russell's Principles of Mathematics seems appropriate here:

In addition to these [indefinable primitives of mathematics], mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth.

I think that much the same can be said of logic, especially math logic. The study of truth is the domain of truth theory. See the SEP article on Truth.

I really can't state with confidence that informal logic is the same, using rather than defining truth. But a quick scan of the SEP article on Informal Logic makes me think that what I've said probably holds of informal logic as well.