def propositional logic: TBD

def predicate logic or first-order logic or first-order predicate logic: a formal language with precise rules of formation. just like computer programming languages(C, Java, etx) (Kunen, 2007:6) (Sakharov).

def proof : 
    def¹ (Kunen, 2007:6): a finite sequence of sentences
                                    written in this language
                                    that obey 
                                        a precisely defined rule of inference.
                                        rules of inference = {modus ponens, ...}
    
    def² (Montalban, 2020): an argument
                                that uses
                                    logical steps
                                to show
                                    that a mathematical statement
                                    follows from certain axioms.


syntactic objects (Kunen, 2007:6):
    - predicate logic sentences
    - formal proofs

def second-order logic: TBD

def higher-order logic: TBD

def: the formal mathematical study of the methods, structure and validity of mathematical deduction and proof (Weisstein).

investigates and classifies the structure of sentences using: - formal systems of inference - analysis/study of arguments in natural language (Wikipedia)

• https://en.wikipedia.org/wiki/Outline_of_logic#Branches_of_logic

• https://en.wikipedia.org/wiki/Outline_of_logic#Branches_of_logic

• Weisstein, Eric W. "Logic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Logic.html
• Kunen, K. 2007. The Foundations of Mathematics.
• Sakharov, A. "First-Order Logic." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/First-OrderLogic.html.
• Montalban, A. 2020. Math 135 -- Intro to Set Theory. https://www.youtube.com/playlist?list=PLjJhPCaCziSQyON7NLc8Ac8ibdm6_iDQf

Q: Question for later research

MYINC: My Insight/Conjecture [Could Be Unoriginal/False and Likely Is]

Disclaimer: This is an ongoing and incomplete project to unpack these concepts and serve as my distributed memory.