ECE 208: Discrete Math 2

Hilbert system rules

Axioms:

• $\vdash (A\implies (B\implies A))$
• $\vdash (A\implies (B\implies C))\implies ((A\implies B)\implies (A\implies C))$
• $\vdash (\neg B\implies\neg A)\implies (A\implies B))$

Inference (MP):

• $\frac{\vdash A, \vdash A\implies B}{\vdash B}$

Derived rules:

• Deduction: $\frac{U\cup \{A\}\vdash B}{U\vdash A\implies B}$
• Contrapositive: $\frac{U\vdash \neg B\implies\neg A}{U\vdash A\implies B}$ (and vice versa)
• Transitivity: $\frac{u\vdash A\implies B, U\vdash B\implies C}{U\vdash A\implies C}$
• Exchange of antecedent: $\frac{U\vdash A\implies (B\implies C)}{U\vdash B\implies (A\implies C)}$
• Double negation: $\frac{U\vdash \neg\neg A}{U\vdash A}$ (and vice versa)
• Reductio ad absurdum: $\frac{U\vdash\neg A\implies false}{U\vdash A}$