I am self-learning William Lawvere's Conceptual Mathematics and Sets For Mathematics. He mentioned subjective and objective logic. I really don't understand it. Do I have to read Hegel's to fully grasp these two books? See this link.
Besides, William Lawvere said in his Categories of space and quantity:
It is my belief that in the next decade and in the next century the technical advances forged by category theorists will be of value to dialectical philosophy, lending precise form with disputable mathematical models to ancient philosophical distinctions such as general vs. particular, objective vs. subjective, being vs. becoming, space vs. quantity, equality vs. difference, quantitative vs. qualitative etc. In turn the explicit attention by mathematicians to such philosophical questions is necessary to achieve the goal of making mathematics (and hence other sciences) more widely learnable and useable. Of course this will require that philosophers learn mathematics and that mathematicians learn philosophy.
Is there any good introductory logic(or philosophical) book covering these aspects(subjective and objective logic) for math students?
Objective logic is the logic intrinsic to a universe of discourse (category). As such, the objective logic of a category can be calculated from the abstract essence(s) in which every object of the category partakes (e.g. singleton set is the abstract essence of the category of sets (in the sense of Isbell's adequacy; http://at.yorku.ca/t/o/p/d/65.htm). The following refs may speak to your question:
Objective Logic of Consciousness
Tools for the Advancement of Objective Logic
Structure and Logic of Conceptual Mind
Subobject Classifier Algorithm
Truth and the Terminal Object