First of all, there is a difference between 'strength' and 'expressiveness'. This is not always unambiguous used in articles and literature.

With 'strength' is usually meant the possibility of a system to proof certain sentences. When comparing two systems for strength, one can limit the comparison to a certain subset of sentences.

Considering your question, I do think that you mean 'expressiveness'.

If you have first order logic + induction scheme + definitions for addition and multiplication, you can express any problem of discrete mathematics.

You need to build a 'pairing' construction and a 'transitive reflexive closure' construction. With those two, you can do anything. With some tricks with addition and multiplication, you can construct both.

You can also opt to start with a pairing operator and closure, and construct addition and multiplication from that.

If you have higher order logic, you don't need the addition and multiplication, because higher order logic allows the construction of pairing and closure in a different way.

So, I think the answer to your question is that you can codify all logics (with finite sentences) with First Order PA (or the question is not clear).

When talking about strength. First order PA + addition + multiplication, can't prove the consistency of itself. You need second order logic, with the ability to have second order formulas as induction hypothesis to prove consistency of first order logic + PA.

Lucas