$\vdash (\lnot \chi \to \lnot \theta) \to (\theta \to \chi) $

I have seen natural deduction proofs that are very complicated, or are written in a way such that I suspect that there is some motivation that I am surely missing. But I can't figure out how to get the intuition/motivation for natural deduction proofs in general.

The following proof is based on a form of natural deduction. Simpler than most, it may help. It uses direct proof (lines 8 and 9) and proof by contradiction (line 6).

Note: '=>' is left associative here, so I am proving the equivalent of [~A => ~B] => [B => A].

Screen shot from my proof checker:

enter image description here

Tags:

Logic

Natural Deduction

Propositional Calculus

Related

Why does an affine transformation $A$ when constrained by $A^TA=\lambda^2I$ result in a similarity transformation? unexplained inequalities from Niven's 1971 proof $\sum 1/p$ diverges Is there a known closed form solution to $\int_0^1\frac{\ln(1+x^{2n})}{1+x^2} \,dx$? Evaluate an absolute monster integral $\int\limits_{0}^{1} \frac{\log(1-x+x^2)}{\sqrt{x}(1+x)}\mathrm{d}x.$ Closed form of $\sum_{n=-\infty}^\infty \frac{(-1)^n}{\sinh (z+n)}$? How to prove an identity by induction Is it correct to have understanding of logic with the use of set theory, to make learning it easier? Proving infinite intersection has only one point How to show that this $2n \times n^2$ matrix has rank $2n-1$? $ \int_c^\infty \frac{e^{-a x^2 + bx}}{x} \, dx $ How can we simplify this integral? $\int{x\frac{f(x)}{\int f(x) dx} dx}$ Prove by induction $2^n \le (n+1)!$

Recent Posts