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Inference Rules
| Legend | |
|---|---|
| α, β, γ | Are well-formed formulas. |
| ψ | Is a predicate (P, Q, R...) |
| c, c´, c´´... | Is an individual constant (a, b, c...) |
| v, v´, v´´ | Is an individual variable (x, y, z...) |
BASIC RULES
| B | ^ | v | → | ↔ | ¬ | ∀ | ∃ | = | ι |
|---|---|---|---|---|---|---|---|---|---|
| (I) |
α β -- α ∧ β
|
α --- αVβ |
┌α |... ┗β --- α→β |
α→β β→α --- α↔β |
┌ ¬α |......... ┗β∧¬β ----- α
|
α --- ∀vα |
α --- ∃vα |
ψϲ -- ∀v(v=ϲ →ψv) |
ψ(ιvα) --- ∃v∀v´ ( α ↔ x=y) |
| (E) |
α ∧ β -- α β
|
┌α |... ┗ γ ┌β |... ┗ γ --- γ |
α→β α --- β |
α↔β ---- α→β β→α |
¬¬α ---- α |
∀vα --- α
|
∃vα --- α |
∀v(v=ϲ →ψv) -- ψϲ |
∃v∀v´ ( Φ ↔ x=y) --- ψ(ιvΦ) |
DERIVED RULES
Derived rules for implication
| RULES | → |
|---|---|
| Transitivity of conditional (Tr→) | α→β β→γ ---- α→γ |
| Modus Tollens (MT) | α→β ¬β ---- ¬α |
| Dilemmas (Dil) | Dil1 α ∨ β α→γ β→γ ---- γ
Dil2 ¬α ∨ ¬β γ→α γ→β ---- ¬γ |
| Premise loading (CrPr) | β ---- α→β |
| Contraposition (Ctrp) | α→β ---- ¬β→¬α |
| Commutation of conditional (Mut →) | α→(β→γ) ---- β→(α→γ) |
| Importation/Exportation (Imp/Exp) | α→(β→γ) ---- α^β→γ |
| Monotonicity (Mon) | α→β ---- α ∧ γ→β |
Derived rules for conjunction / disjunction
| RULES | ^ | v |
|---|---|---|
| Disjunctive Syllogism (DS) | α ∨ β ¬α ---- β
α ∨ β ¬β ---- α |
|
| Idempotence (Idp ∧ / Idp v) | α ∧ α ----- α |
α ∨ α ---- α |
| Absorption (Abs ^/Abs v) | α ∧ (α ∨ β) ---- α |
α ∨ (α ∧ β) ---- α |
| Commutativity (Comm ^) | α ∧ β ---- β ∧ α |
α ∨ β ---- β ∨ α |
| Associativity (Assoc ^/Assoc v) | (α ∧ β) ∧ γ ---- α ∧ (β ∧ γ) |
(α ∨ β) ∨ γ ---- α ∨ (β ∨ γ) |
| Distribution (Dist ^/Dist v) | α ∧ (β ∨ γ) ---- (α ∧ β) ∨ (α ∧ γ) |
α ∨ (β ∧ γ) ---- (α ∨ β) ∧ (α ∨ γ) |
Derived rules for quantifiers
| RULES | ∀ | ∃ | ||
|---|---|---|---|---|
| Negation of Universal or Existential Quantifier (Neg Gen or Neg Par) | ¬∀vα ---- ∃v¬α |
¬∃vα ---- ∀v¬α |
||
| Quantifier descent (Des Quant) | ∀vα ---- ∃vα |
---- | ||
| Variable mutation (Mut Var) | ∀vα ---- ∀v´α |
∃vα ---- ∃v´α |
||
| Contraction of Universal or Existential (Contract Gen Disj or Contract Part Cond) | ∀vα ∨ ∀vβ ---- ∀v(α ∨ β) |
∃vα → ∃vβ ---- ∃v(α → β) |
||
Quantifier permutations (Perm Gen) |
∀v∀v´α ----- ∀v´∀vα |
∃v∃v´α ----- ∃v´∃vα |
||
∃v∀v´α ----- ∀v´∃vα |
||||
| Distribution of Universal or Existential in conjunction (Dist Gen ∧ / Dist Part ^) | ∀v(α ∧ β) ---- ---- ∀vα ∧ ∀vβ |
∃v(α ∧ β)
---- ∃vα ∧ ∃vβ |
||
| Distribution of Existential in disjunction (Dist Part v) | ---- | ∃v(α ∨ β) ---- ---- ∃vα ∨ ∃vβ |
||
| Distribution of Universal and Existential in conditional (Dist Gen →/Dist Part →) | ∀v(α→β) ----
∀vα→∀vβ |
∃v(α→β) ---- ---- ∀vα→∃vβ |
||
| Distribution of Universal in Biconditional (Dist Gen ↔) | ∀v(α↔β)
----
∀vα↔∀vβ |
|||
Conditional distribution of universal for conjunction, disjunction, antecedent and consequent. (Dist Gen ^/v/Antec/Consec) |
α ∧ ∀vβ ---- ∀v(α ^β) |
α ∨ ∀vβ ---- ∀v(α vβ) |
∀vβ → α --- ∃v(β →α) |
α→∀vβ ---- ∀v(α→β) |
Conditional distribution of existential for conjunction, disjunction, antecedent and consequent. (Dist Part) ^/v/Antec/Consec) |
α ∧ ∃vβ ---- ∃v(α ^β) |
α ∨ ∃vβ ---- ∃v(α vβ) |
∃vβ → α --- ∀v(β →α) |
α→∃vβ ---- ∃v(α→β) |
Derived rules for identity
| RULES | = |
|---|---|
| Leibniz1 | c=c' ψc ---- ψc' |
| Leibniz2 | c=c' ψc´ ---- ψc
|
| Leibniz3 | ψc ¬ψc´ ---- c≠c´ |
| Leibniz4 | ¬ψc ψc´ ---- c≠c´ |
| Reflexivity of identity (Refl =) | ---- c=c' |
| Symmetry of identity (Sym =) | c = c´ ---- c´= c |
| Transitivity of identity (Tr =) | c=c' c´=c´´ ---- c = c´´ |
| Indiscernibility (Indiscer) | c=c' ---- ψc↔ψc´ |
| Euclid | c=c' ---- fc=fc´ |
Definition rules
| RULES | Connective 1 | Connective 2 |
|---|---|---|
| DM ^/v v/v | ¬(α ∧ β) ---- ¬α ∨ ¬ β |
¬(α ∨ β) ---- ¬α ∧ ¬ β |
| Definition ^/v v/^ | α ∧ β ---- ¬(¬α ∨ ¬ β) |
α ∨ β ---- ¬(¬α ∧ ¬ β) |
| Definition ^/→ v/→ | α ∧ β ---- ¬(α → ¬ β) |
α ∨ β ---- ¬α → β |
| Definition ∃/∀ ∀/∃ | ∀vα ---- ¬∃v¬α |
∃vα ---- ¬∀v¬α |
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