= Mathematical logic
{wiki}
Mathematical logic studies <formal languages>, <formal proofs>, <model theory>, <computability theory>, and the <foundations of mathematics>. It makes the rules of reasoning explicit enough that reasoning itself becomes an object of <mathematics>. <Proof theory> studies deductions; <model theory> studies interpretations in which statements hold.
Three questions organize the subject: is an expression a well-formed <first-order formula>, is it true in a specified <first-order structure>, and does it have a <formal proof> from specified <axioms>? These questions are distinct. For example, $\exists n\,(n+n=1)$ is a <first-order sentence> in an appropriate <first-order language>; it is false in the <natural numbers> but true in the <rational numbers>, with $n=1/2$.
The beauty of <mathematical logic> is that the same <logical connectives> and rules of <universal quantification> and <existential quantification> organize arguments across <mathematics>. <Classical propositional logic> uses two truth values; <Intuitionistic propositional logic> interprets <logical implication> through constructions on proofs, without unrestricted <law of excluded middle>. This proof-based framework is developed in https://www.cs.cmu.edu/~fp/courses/15317-s23/lectures/02-natded.pdf[Frank Pfenning, Natural Deduction, sections 1–3].
Back to article page
