Module: Sentential logic
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To understand the actual world as it is, not as we should wish it to be, is the beginning of wisdom.
- Bertrand Russell
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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 (φ→ψ), depending on everything ψ depends on except φ.
→E (Conditional Elimination or Modus Ponens)
If you have derived (φ→ψ) and φ, you can write down ψ, depending on everything (φ→ψ) 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 (φ∨ψ)
or (ψ∨φ), depending on everything φ depends on. (ψ is any SL wff.)
∨E (Disjunction Elimination or Disjunctive Syllogism)
If you have derived (φ∨ψ) and ~ψ,you can write down φ,
depending on everything (φ∨ψ) and ~ψ depend on.
Also, if you have derived (φ∨ψ) and ~φ, you can write down ψ, depending on everything (φ∨ψ) and ~φ depend on.
PC (Proof by Cases)
If you have derived (φ∨ψ) and (φ→α)and (ψ→β), then you can write down (α
∨β),depending on everything (φ∨ψ) and (φ→α) and (ψ→β) depend on.
↔I (Biconditional Introduction)
If you have derived ((φ→ψ)&(ψ→φ)),
you can write down (φ↔ψ), depending on everything ((φ→ψ)&(ψ→φ)) depends on.
↔E (Biconditional Elimination)
If you have derived (φ↔ψ), you can write down ((φ→ψ)&(ψ→φ)) depending on everything (φ
↔ψ) depends on.