Natural Deduction Rules for Sentential Logic
(Revised 5 January 2011)

Note: in the following rules, the greek letters "φ" and
"ψ" are names of SL wffs.


A (Rule of Assumption)
You can write down any SL wff, depending on itself.


&I (Conjunction Introduction)
If you have derived φ and ψ,
you can write down (φ&ψ),
depending on everything φ and ψ depend on.


&E (Conjunction Elimination)
If you have derived (φ&ψ),
you can write down φ or ψ,
depending on everything (φ&ψ) depends on.


I (Conditional Introduction)
If you have assumed φ, and you have derived ψ,
you can write down (φψ),
depending on everything ψ depends on except φ.


E (Conditional Elimination or Modus Ponens)
If you have derived (φψ) and φ,
you can write down ψ,
depending on everything (φψ) and φ depend on.


~I (Negation Introduction)
If you have assumed ψ, and you have derived (φ&~φ),
then you can write down ~ψ,
depending on everything (φ&~φ) depends on except ψ.


~E (Negation Elimination)
If you have assumed ~ψ, and you have derived (φ&~φ),
then you can write down ψ,
depending on everything (φ&~φ) depends on except ~ψ.


I (Disjunction Introduction)
If you have derived φ,
you can write down (φψ) or (ψφ),
depending on everything φ depends on.
(ψ is any SL wff.)


E (Disjunction Elimination or Disjunctive Syllogism)
If you have derived (φψ) and ~ψ,
you can write down φ,
depending on everything (φψ) and ~ψ depend on.

If you have derived (φψ) and ~φ,
you can write down ψ,
depending on everything (φψ) and ~φ depend on.


PC (Proof by Cases)
If you have derived (φψ) and (φα) and (ψβ),
then you can write down (αβ),
depending on everything (φψ) and
α) and (ψβ) depend on.


I (Biconditional Introduction)
If you have derived ((φψ)&(ψφ)),
you can write down (φψ),
depending on everything ((φψ)&(ψφ)) depends on.


E (Biconditional Elimination)
If you have derived (φψ),
you can write down ((φψ)&(ψφ))
depending on everything (φψ) depends on.