Propositional Tautologies

Proposition 1: ${\alpha}{\;\limplies\;}{\alpha}$

Quine Alternatives:

• ${\alpha}{\;\limplies\;}{\alpha}$ $[{\alpha}:={\ltrue}]$

  Simplification

  • ${\ltrue}{\;\limplies\;}{\ltrue}$
  • ${\ltrue}$

  Reduced to ${\ltrue}$.

• ${\alpha}{\;\limplies\;}{\alpha}$ $[{\alpha}:={\lfalse}]$

  Simplification

  • ${\lfalse}{\;\limplies\;}{\lfalse}$
  • ${\ltrue}$

  Reduced to ${\ltrue}$.

$\Box$ Proposition 1

Proposition 2: ${\alpha}{\;\limplies\;}{\beta}{\;\limplies\;}{\alpha}$

Quine Alternatives:

• ${\alpha}{\;\limplies\;}{\beta}{\;\limplies\;}{\alpha}$ $[{\alpha}:={\ltrue}]$

  Simplification

  • ${\ltrue}{\;\limplies\;}{\beta}{\;\limplies\;}{\ltrue}$
  • ${\beta}{\;\limplies\;}{\ltrue}$
  • ${\ltrue}$

  Reduced to ${\ltrue}$.

• ${\alpha}{\;\limplies\;}{\beta}{\;\limplies\;}{\alpha}$ $[{\alpha}:={\lfalse}]$

  Simplification

  • ${\lfalse}{\;\limplies\;}{\beta}{\;\limplies\;}{\lfalse}$
  • ${\ltrue}$

  Reduced to ${\ltrue}$.

$\Box$ Proposition 2

Proposition 3: ${\alpha}{\;\limplies\;}({\alpha}{\;\limplies\;}{\beta}){\;\limplies\;}{\beta}$

Quine Alternatives:

• ${\alpha}{\;\limplies\;}({\alpha}{\;\limplies\;}{\beta}){\;\limplies\;}{\beta}$ $[{\alpha}:={\ltrue}]$

  • ${\ltrue}{\;\limplies\;}({\ltrue}{\;\limplies\;}{\beta}){\;\limplies\;}{\beta}$
  • $({\ltrue}{\;\limplies\;}{\beta}){\;\limplies\;}{\beta}$
  • ${\beta}{\;\limplies\;}{\beta}$

  Substitution instance of Proposition 1$[{\alpha}{\;\limplies\;}{\alpha}]$ at

  • ${\alpha} := {\beta}$

• ${\alpha}{\;\limplies\;}({\alpha}{\;\limplies\;}{\beta}){\;\limplies\;}{\beta}$ $[{\alpha}:={\lfalse}]$

  Simplification

  • ${\lfalse}{\;\limplies\;}({\lfalse}{\;\limplies\;}{\beta}){\;\limplies\;}{\beta}$
  • ${\ltrue}$

  Reduced to ${\ltrue}$.

$\Box$ Proposition 3