Guest access

pasteur

pasteur

Points:	10.563
Rank:	116
Opponents defeated:	19.303 (174.)
Tribe:	OGs

Villages (3)

Coordinates

Points

01 Euler's theorem

467|460

6.336

02 Lagrangian method

470|457

2.634

03 Lowenheim-Skolem theorem

467|465

1.593

Personal text

Intuitionistic logic can be defined using the following Hilbert-style calculus. This is similar to a way of axiomatizing classical propositional logic.

In propositional logic, the inference rule is modus ponens

MP: from ϕ → ψ and ϕ infer ψ
and the axioms are

THEN-1: ψ → ( ϕ → ψ )
THEN-2: ( χ → ( ϕ → ψ ) ) → ( ( χ → ϕ ) → ( χ → ψ ) )
AND-1: ϕ ∧ χ → ϕ
AND-2: ϕ ∧ χ → χ
AND-3: ϕ → ( χ → ( ϕ ∧ χ ) )
OR-1: ϕ → ϕ ∨ χ
OR-2: χ → ϕ ∨ χ
OR-3: ( ϕ → ψ ) → ( ( χ → ψ ) → ( ( ϕ ∨ χ ) → ψ ) )
FALSE: ⊥ → ϕ

To make this a system of first-order predicate logic, the generalization rules

∀-GEN: from ψ → ϕ infer ψ → ( ∀ x ϕ ), if x {\displaystyle x} is not free in ψ {\displaystyle \psi }
∃-GEN: from ϕ → ψ infer ( ∃ x ϕ ) → ψ if x is not free in ψ

are added, along with the axioms

PRED-1: ( ∀ x ϕ ( x ) ) → ϕ ( t ), if the term t is free for substitution for the variable x in ϕ (i.e., if no occurrence of any variable in t becomes bound in ϕ ( t ))
PRED-2: ϕ ( t ) → ( ∃ x ϕ ( x ) ), with the same restriction as for PRED-1

Combat achievements