Propositional Tautologies

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

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: $\proves\;{\alpha}\:\limplies\:{\beta}\:\limplies\:{\alpha}$

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: $\proves\;{\alpha}\:\limplies\:({\alpha}\:\limplies\:{\beta})\:\limplies\:{\beta}$

Quine Alternatives:

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

  Simplification

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

  Instance of Proposition 1: ${\alpha}\:\limplies\:{\alpha}$.

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

  Simplification

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

  Reduced to ${\ltrue}$.

$\Box$ Proposition 3

Proposition 4: $\proves\;({\alpha}\:\limplies\:{\beta})\:\limplies\:({\alpha}\:\limplies\:{\beta}\:\limplies\:{\gamma})\:\limplies\:{\alpha}\:\limplies\:{\gamma}$

Quine Alternatives:

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

  Simplification

  • $({\ltrue}\:\limplies\:{\beta})\:\limplies\:({\ltrue}\:\limplies\:{\beta}\:\limplies\:{\gamma})\:\limplies\:{\ltrue}\:\limplies\:{\gamma}$
  • ${\beta}\:\limplies\:({\beta}\:\limplies\:{\gamma})\:\limplies\:{\gamma}$

  Instance of Proposition 3: ${\alpha}\:\limplies\:({\alpha}\:\limplies\:{\beta})\:\limplies\:{\beta}$.

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

  Simplification

  • $({\lfalse}\:\limplies\:{\beta})\:\limplies\:({\lfalse}\:\limplies\:{\beta}\:\limplies\:{\gamma})\:\limplies\:{\lfalse}\:\limplies\:{\gamma}$
  • ${\ltrue}\:\limplies\:{\ltrue}\:\limplies\:{\ltrue}$
  • ${\ltrue}\:\limplies\:{\ltrue}$
  • ${\ltrue}$

  Reduced to ${\ltrue}$.

$\Box$ Proposition 4

Proposition 5: $\proves\;({\alpha}\:\limplies\:{\beta})\:\limplies\:({\beta}\:\limplies\:{\gamma})\:\limplies\:{\alpha}\:\limplies\:{\gamma}$

Quine Alternatives:

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

  Simplification

  • $({\ltrue}\:\limplies\:{\beta})\:\limplies\:({\beta}\:\limplies\:{\gamma})\:\limplies\:{\ltrue}\:\limplies\:{\gamma}$
  • ${\beta}\:\limplies\:({\beta}\:\limplies\:{\gamma})\:\limplies\:{\gamma}$

  Instance of Proposition 3: ${\alpha}\:\limplies\:({\alpha}\:\limplies\:{\beta})\:\limplies\:{\beta}$.

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

  Simplification

  • $({\lfalse}\:\limplies\:{\beta})\:\limplies\:({\beta}\:\limplies\:{\gamma})\:\limplies\:{\lfalse}\:\limplies\:{\gamma}$
  • ${\ltrue}\:\limplies\:({\beta}\:\limplies\:{\gamma})\:\limplies\:{\ltrue}$
  • $({\beta}\:\limplies\:{\gamma})\:\limplies\:{\ltrue}$
  • ${\ltrue}$

  Reduced to ${\ltrue}$.

$\Box$ Proposition 5

Proposition 6: $\proves\;{\alpha}\,\lor\,{\beta}\:\limplies\:{\lnot}{\alpha}\:\limplies\:{\beta}$

Quine Alternatives:

• ${\alpha}\,\lor\,{\beta}\:\limplies\:{\lnot}{\alpha}\:\limplies\:{\beta}$ [${\alpha}$:=${\ltrue}$]

  Simplification

  • ${\ltrue}\,\lor\,{\beta}\:\limplies\:{\lnot}{\ltrue}\:\limplies\:{\beta}$
  • ${\ltrue}\:\limplies\:{\lfalse}\:\limplies\:{\beta}$
  • ${\lfalse}\:\limplies\:{\beta}$
  • ${\ltrue}$

  Reduced to ${\ltrue}$.

• ${\alpha}\,\lor\,{\beta}\:\limplies\:{\lnot}{\alpha}\:\limplies\:{\beta}$ [${\alpha}$:=${\lfalse}$]

  Simplification

  • ${\lfalse}\,\lor\,{\beta}\:\limplies\:{\lnot}{\lfalse}\:\limplies\:{\beta}$
  • ${\beta}\:\limplies\:{\ltrue}\:\limplies\:{\beta}$
  • ${\beta}\:\limplies\:{\beta}$

  Instance of Proposition 1: ${\alpha}\:\limplies\:{\alpha}$.

$\Box$ Proposition 6

Proposition 7: $\proves\;({\alpha}\:\limplies\:{\beta})\:\limplies\:({\lnot}{\alpha}\:\limplies\:{\beta})\:\limplies\:{\beta}$

Quine Alternatives:

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

  Simplification

  • $({\alpha}\:\limplies\:{\ltrue})\:\limplies\:({\lnot}{\alpha}\:\limplies\:{\ltrue})\:\limplies\:{\ltrue}$
  • ${\ltrue}\:\limplies\:{\ltrue}$
  • ${\ltrue}$

  Reduced to ${\ltrue}$.

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

  Simplification

  • $({\alpha}\:\limplies\:{\lfalse})\:\limplies\:({\lnot}{\alpha}\:\limplies\:{\lfalse})\:\limplies\:{\lfalse}$
  • ${\lnot}{\alpha}\:\limplies\:{\lnot}({\lnot}{\alpha}\:\limplies\:{\lfalse})$
  • ${\lnot}{\alpha}\:\limplies\:{\lnot}{\lnot}{\lnot}{\alpha}$
  • ${\lnot}{\alpha}\:\limplies\:{\lnot}{\alpha}$

  Instance of Proposition 1: ${\alpha}\:\limplies\:{\alpha}$.

$\Box$ Proposition 7

Proposition 8: $\proves\;({\lnot}{\alpha}\:\limplies\:{\beta})\:\limplies\:({\lnot}{\alpha}\:\limplies\:{\lnot}{\beta})\:\limplies\:{\alpha}$

Quine Alternatives:

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

  Simplification

  • $({\lnot}{\ltrue}\:\limplies\:{\beta})\:\limplies\:({\lnot}{\ltrue}\:\limplies\:{\lnot}{\beta})\:\limplies\:{\ltrue}$
  • $({\lfalse}\:\limplies\:{\beta})\:\limplies\:{\ltrue}$
  • ${\ltrue}$

  Reduced to ${\ltrue}$.

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

  Simplification

  • $({\lnot}{\lfalse}\:\limplies\:{\beta})\:\limplies\:({\lnot}{\lfalse}\:\limplies\:{\lnot}{\beta})\:\limplies\:{\lfalse}$
  • $({\ltrue}\:\limplies\:{\beta})\:\limplies\:{\lnot}({\lnot}{\lfalse}\:\limplies\:{\lnot}{\beta})$
  • ${\beta}\:\limplies\:{\lnot}({\ltrue}\:\limplies\:{\lnot}{\beta})$
  • ${\beta}\:\limplies\:{\lnot}{\lnot}{\beta}$
  • ${\beta}\:\limplies\:{\beta}$

  Instance of Proposition 1: ${\alpha}\:\limplies\:{\alpha}$.

$\Box$ Proposition 8

Proposition 9: $\proves\;{\alpha}\,\lor\,{\beta}\:\limplies\:({\alpha}\:\limplies\:{\gamma})\:\limplies\:({\beta}\:\limplies\:{\gamma})\:\limplies\:{\gamma}$

Quine Alternatives:

• ${\alpha}\,\lor\,{\beta}\:\limplies\:({\alpha}\:\limplies\:{\gamma})\:\limplies\:({\beta}\:\limplies\:{\gamma})\:\limplies\:{\gamma}$ [${\gamma}$:=${\ltrue}$]

  Simplification

  • ${\alpha}\,\lor\,{\beta}\:\limplies\:({\alpha}\:\limplies\:{\ltrue})\:\limplies\:({\beta}\:\limplies\:{\ltrue})\:\limplies\:{\ltrue}$
  • ${\alpha}\,\lor\,{\beta}\:\limplies\:{\ltrue}\:\limplies\:{\ltrue}$
  • ${\alpha}\,\lor\,{\beta}\:\limplies\:{\ltrue}$
  • ${\ltrue}$

  Reduced to ${\ltrue}$.

• ${\alpha}\,\lor\,{\beta}\:\limplies\:({\alpha}\:\limplies\:{\gamma})\:\limplies\:({\beta}\:\limplies\:{\gamma})\:\limplies\:{\gamma}$ [${\gamma}$:=${\lfalse}$]

  Simplification

  • ${\alpha}\,\lor\,{\beta}\:\limplies\:({\alpha}\:\limplies\:{\lfalse})\:\limplies\:({\beta}\:\limplies\:{\lfalse})\:\limplies\:{\lfalse}$
  • ${\alpha}\,\lor\,{\beta}\:\limplies\:{\lnot}{\alpha}\:\limplies\:{\lnot}({\beta}\:\limplies\:{\lfalse})$
  • ${\alpha}\,\lor\,{\beta}\:\limplies\:{\lnot}{\alpha}\:\limplies\:{\lnot}{\lnot}{\beta}$
  • ${\alpha}\,\lor\,{\beta}\:\limplies\:{\lnot}{\alpha}\:\limplies\:{\beta}$

  Instance of Proposition 6: ${\alpha}\,\lor\,{\beta}\:\limplies\:{\lnot}{\alpha}\:\limplies\:{\beta}$.

$\Box$ Proposition 9

Proposition 10: $\proves\;{\alpha}\,\lor\,{\beta}\,\lor\,{\gamma}\:\limplies\:({\alpha}\:\limplies\:{\delta})\:\limplies\:({\beta}\:\limplies\:{\delta})\:\limplies\:({\gamma}\:\limplies\:{\delta})\:\limplies\:{\delta}$

Quine Alternatives:

• ${\alpha}\,\lor\,{\beta}\,\lor\,{\gamma}\:\limplies\:({\alpha}\:\limplies\:{\delta})\:\limplies\:({\beta}\:\limplies\:{\delta})\:\limplies\:({\gamma}\:\limplies\:{\delta})\:\limplies\:{\delta}$ [${\delta}$:=${\ltrue}$]

  Simplification

  • ${\alpha}\,\lor\,{\beta}\,\lor\,{\gamma}\:\limplies\:({\alpha}\:\limplies\:{\ltrue})\:\limplies\:({\beta}\:\limplies\:{\ltrue})\:\limplies\:({\gamma}\:\limplies\:{\ltrue})\:\limplies\:{\ltrue}$
  • ${\alpha}\,\lor\,{\beta}\,\lor\,{\gamma}\:\limplies\:{\ltrue}\:\limplies\:{\ltrue}\:\limplies\:{\ltrue}$
  • ${\alpha}\,\lor\,{\beta}\,\lor\,{\gamma}\:\limplies\:{\ltrue}\:\limplies\:{\ltrue}$
  • ${\alpha}\,\lor\,{\beta}\,\lor\,{\gamma}\:\limplies\:{\ltrue}$
  • ${\ltrue}$

  Reduced to ${\ltrue}$.

• ${\alpha}\,\lor\,{\beta}\,\lor\,{\gamma}\:\limplies\:({\alpha}\:\limplies\:{\delta})\:\limplies\:({\beta}\:\limplies\:{\delta})\:\limplies\:({\gamma}\:\limplies\:{\delta})\:\limplies\:{\delta}$ [${\delta}$:=${\lfalse}$]

  Simplification

  • ${\alpha}\,\lor\,{\beta}\,\lor\,{\gamma}\:\limplies\:({\alpha}\:\limplies\:{\lfalse})\:\limplies\:({\beta}\:\limplies\:{\lfalse})\:\limplies\:({\gamma}\:\limplies\:{\lfalse})\:\limplies\:{\lfalse}$
  • ${\alpha}\,\lor\,{\beta}\,\lor\,{\gamma}\:\limplies\:{\lnot}{\alpha}\:\limplies\:{\lnot}{\beta}\:\limplies\:{\lnot}({\gamma}\:\limplies\:{\lfalse})$
  • ${\alpha}\,\lor\,{\beta}\,\lor\,{\gamma}\:\limplies\:{\lnot}{\alpha}\:\limplies\:{\lnot}{\beta}\:\limplies\:{\lnot}{\lnot}{\gamma}$
  • ${\alpha}\,\lor\,{\beta}\,\lor\,{\gamma}\:\limplies\:{\lnot}{\alpha}\:\limplies\:{\lnot}{\beta}\:\limplies\:{\gamma}$

  Quine Alternatives:

  • ${\alpha}\,\lor\,{\beta}\,\lor\,{\gamma}\:\limplies\:({\alpha}\:\limplies\:{\delta})\:\limplies\:({\beta}\:\limplies\:{\delta})\:\limplies\:({\gamma}\:\limplies\:{\delta})\:\limplies\:{\delta}$ [${\alpha}$:=${\ltrue}$]

    Simplification

    • ${\ltrue}\,\lor\,{\beta}\,\lor\,{\gamma}\:\limplies\:{\lnot}{\ltrue}\:\limplies\:{\lnot}{\beta}\:\limplies\:{\gamma}$
    • ${\ltrue}\,\lor\,{\gamma}\:\limplies\:{\lfalse}\:\limplies\:{\lnot}{\beta}\:\limplies\:{\gamma}$
    • ${\ltrue}\:\limplies\:{\ltrue}$
    • ${\ltrue}$

    Reduced to ${\ltrue}$.

  • ${\alpha}\,\lor\,{\beta}\,\lor\,{\gamma}\:\limplies\:({\alpha}\:\limplies\:{\delta})\:\limplies\:({\beta}\:\limplies\:{\delta})\:\limplies\:({\gamma}\:\limplies\:{\delta})\:\limplies\:{\delta}$ [${\alpha}$:=${\lfalse}$]

    Simplification

    • ${\lfalse}\,\lor\,{\beta}\,\lor\,{\gamma}\:\limplies\:{\lnot}{\lfalse}\:\limplies\:{\lnot}{\beta}\:\limplies\:{\gamma}$
    • ${\beta}\,\lor\,{\gamma}\:\limplies\:{\ltrue}\:\limplies\:{\lnot}{\beta}\:\limplies\:{\gamma}$
    • ${\beta}\,\lor\,{\gamma}\:\limplies\:{\lnot}{\beta}\:\limplies\:{\gamma}$

    Instance of Proposition 6: ${\alpha}\,\lor\,{\beta}\:\limplies\:{\lnot}{\alpha}\:\limplies\:{\beta}$.

$\Box$ Proposition 10