** Module: Predicate logic**

- PL00. Introduction
- PL01. Singular terms
- PL02. Variables and predicates
- PL03. Quantifiers 1
- PL04. Quantifiers 2
- PL05. Well-formed formula
- PL06. Interpretation
- PL07. Domain
- PL08. Validity
- PL09. Other valid sequents
- PL10. Derivations 1
- PL11. Derivation rules
- PL12. Derivations 2
- PL13. Soundness and completeness

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- Erasmus

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Here is a summary of the Natural Deduction Rules for MPL.

**∀E (Universal Quantifier Elimination) **

For any variable v and name c,
if you have derived ∀vφ,
then you can write down φv/c,
depending on everything ∀vφ depends on.

**∃I (Existential Quantifier Introduction) **

For any variable v and name c,
if you have derived φv/c,
and ∃vφ is a well-formed formula of MPL,
then you can write down ∃vφ,
depending on everything φv/c depends on.

**∀I (Universal Quantifier Introduction) **

For any variable v and name c,
if you have derived φv/c, and c does not occur in φ,
and c does not occur in anything φv/c depends on,
and ∀vφ is a well-formed formula of MPL,
then you can write down ∀vφ,
depending on everything φv/c depends on.

**∃E (Existential Quantifier Elimination) **

For any variable v and name c,
if you have derived ∃vφ, assumed φv/c, and derived ψ,
and c does not occur in ψ, φ, or anything ψ depends on (except φv/c),
then you can write down ψ a second time, depending on everything
the first ψ depends on (except the assumption φv/c) together with
everything ∃vφ depends on.

**A (Rule of Assumption) **

You can write down any MPL 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 MPL 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.