This is also somewhat in reply to your elaboration in this comment. Just some data points:
In regards to this topic of proof, and more generally to the topic of formal science, I have found logic a very useful subject. For one, you can leverage your verbal reasoning ability, and begin by conceiving of it as a symbolization of natural language, which I find for myself and many others is far more convenient than, say, a formal science that requires more spatial reasoning or abstract pattern recognition. Later, the point that formal languages are languages in their own right is driven home, and you can do away with this conceptual bridge.
Logic also has helped me to conceive of formal problems as a continuum of difficulty of proof, rather than proofs and non-proofs. That is, when you read a math textbook, sometimes you are instructed to Solve, sometimes to Evaluate, sometimes to Graph; and then there is the dreaded Show That X or Prove That X! In a logic textbook, almost all exercises require a proof of validity, and you move up over time, deriving new inference rules from old, and moving onto metalogical theorems. Later returning to books about mathematical proof, I found things much less intimidating. I found that proof is not a realm forbidden to those lacking an innate ability to prove; you must work your way upwards as in all things.
Furthermore, in regards to this:
Even the math with simple foundations has surprising results with complicated proofs that require precise understanding.
In my opinion, very significant and complex results in logic are arrived at quite early in comparison to the significance of, and effort invested in, results in other fields of formal science.
And in regards to this:
I think this makes it all the more educational when a surprising result is proven, because there is less room for a beginner to wonder whether the result is an artifact of the funny formalish stuff.
I have found that in continuous mathematics I have walked away from proofs with a feeling best expressed as, “If you say so,” as opposed to discrete mathematics and logic, where it’s more like, “Why, of course!”
This is also somewhat in reply to your elaboration in this comment. Just some data points:
In regards to this topic of proof, and more generally to the topic of formal science, I have found logic a very useful subject. For one, you can leverage your verbal reasoning ability, and begin by conceiving of it as a symbolization of natural language, which I find for myself and many others is far more convenient than, say, a formal science that requires more spatial reasoning or abstract pattern recognition. Later, the point that formal languages are languages in their own right is driven home, and you can do away with this conceptual bridge.
Logic also has helped me to conceive of formal problems as a continuum of difficulty of proof, rather than proofs and non-proofs. That is, when you read a math textbook, sometimes you are instructed to Solve, sometimes to Evaluate, sometimes to Graph; and then there is the dreaded Show That X or Prove That X! In a logic textbook, almost all exercises require a proof of validity, and you move up over time, deriving new inference rules from old, and moving onto metalogical theorems. Later returning to books about mathematical proof, I found things much less intimidating. I found that proof is not a realm forbidden to those lacking an innate ability to prove; you must work your way upwards as in all things.
Furthermore, in regards to this:
In my opinion, very significant and complex results in logic are arrived at quite early in comparison to the significance of, and effort invested in, results in other fields of formal science.
And in regards to this:
I have found that in continuous mathematics I have walked away from proofs with a feeling best expressed as, “If you say so,” as opposed to discrete mathematics and logic, where it’s more like, “Why, of course!”