Frederic FitchFrederic B. Fitch was the Sterling Professor of Logic at Yale. Among those attending his classes on propositional logic based on the great Principia Mathematica of Bertrand Russell and Alfred North Whitehead was Warren McCulloch, who went on to establish the modern idea that the brain is a logical machine equivalent to a digital computer (the ideal Turing Machine). Despite the fact that information flows along neurons in the brain, the neural network is not a computer network, brain processes are not algorithms, there is no central processing unit (CPU) or even distributed parallel processing. Very simply, man is not a machine and the brain is not a computer. In general, mathematicians in general and logicians in particular should know that logic is a tool used in scientific reasoning, but it does not control the physical world. Fitch's 1952 textbook Symbolic Logic, An Introduction was important in the history of modern debates about the "necessity of identity." Fitch was Ruth Barcan (Marcus)' thesis adviser. In 1947, Marcus wrote an article on "The Identity of Individuals " asserting the "necessity of identity." Her work was written in the dense expressions of symbolic logic, with little explanation. We present it for historical completeness,
2.33*. ⊢ (β1I(β2) ≡ (β1Im(β2).Five years later, Fitch published his book, Symbolic Logic, which contained the simplest proof ever of the necessity of identity, by the simple mathematical substitution of b for a in the necessity of self-identity statement.
23.4Clearly this is mathematically and logically sound. Fitch substitutes b from (1), for a in the modal context of (2). This would be fine if these are just equations. But as Barcan Marcus knew very well from C.S.Lewis's work on strict implication, substitutivity in statements also requires that the substitution is intensionally meaningful. In the sense that b is actually just a, substituting b is equivalent to keeping a there, a tautology, something with no new information. To be informative and prove the necessary truth of the new statement, we must know more about b, for example, that its intrinsic information in b is identical to that of a. Extrinsic information can differ. Normal | Teacher | Scholar
(1) a = b,
(2) ◻[a = a],
then (3) ◻[a = b], by identity elimination. (p.164)