Karel Lambert

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, ${\displaystyle \forall x\,\phi x\rightarrow \phi y}$, with universal specification ${\displaystyle (\forall x\,\phi x\land E!y\,\phi y)\rightarrow \phi z}$. 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 ${\displaystyle \forall x\,(Mx\rightarrow Lx)}$ and ${\displaystyle \forall x\,Ux}$, which in Principia Mathematica entail ${\displaystyle \exists x\,(Mx\land Lx)}$ and ${\displaystyle \exists x\,Ux}$, 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.

• Philosophical Applications of Free Logic, New York: Oxford University Press, 1991, "A Theory of Definite Descriptions", pp. 17–27, details an account of Russell's Theory of Descriptions in free logic. In the process, he demonstrates how a formulation from Hintikka allows for a contradiction by a correlate in logic to Russell's Paradox. He introduces the predicate ${\displaystyle (\lambda x)(\phi x\land \neg \phi x)}$
• Free Logic. Selected Essays, Cambridge University Press, 2003.

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