Karel Lambert

Lambert's Law

Lambert's Law is the major principle in any free definite description theory that says: For all x, x = the y (A) if and only if (A(x/y) & for all y (if A then y = x)). Free logic itself is an adjustment of a given standard predicate logic such as to relieve it of existential assumptions, and so make it a free logic. Taking Bertrand Russell's predicate logic in his Principia Mathematica as standard, one replaces universal instantiation, ∀ x ϕ x → ϕ y , with universal specification ( ∀ x ϕ x ∧ E ! y ϕ y ) → ϕ z . Thus universal statements, like "All men are mortal," or "Everything is a unicorn," do not presuppose that there are men or that there is anything. These would be symbolized, with the appropriate predicates, as ∀ x ( M x → L x ) and ∀ x U x , which in Principia Mathematica entail ∃ x ( M x ∧ L x ) and ∃ x U x , but not in free logic. The truth of these last statements, when used in a free logic, depend on the domain of quantification, which may be the null set.

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