230 Logic as a Tool mathematics of the continuum He also conducted important work in several other branches of pure and applied mathematics and statistics Notably, as early as in 1886 – more than 50 years before the first digital computers were constructed – he showed how Boolean logical operations could be carried out by electrical switching circuits Peirce’s personal life and professional career were quite troubled From early age he suffered from a nervous condition now called “facial neuralgia” which made him appear unfriendly and also caused sudden outbursts of bad temper Together with some other features of his character, this contributed to the fact that he never held a stable employment; the only academic job he had was a non-tenure position as lecturer in logic at the Johns Hopkins University However, he was fired from this post in 1884 after a scandal following the discovery that, while still legally married pending divorce, he had an affair with another woman (of unknown origin) before they married later and lived together to his end Peirce spent the last 20–30 years of his life in utmost poverty and constant debt, and died destitute While Peirce was largely unrecognized and rejected during his life, gradually he received the recognition he deserved after his death Russell and Whitehead were not aware of his work when they published Principia Mathematica in 1910 and did not mention him there, but in 1959 Russell wrote “Beyond doubt [ ] he was one of the most original minds of the later nineteenth century, and certainly the greatest American thinker ever.” Much earlier in 1918, the logician C I Lewis wrote “The contributions of C.S Peirce to symbolic logic are more numerous and varied than those of any other writer – at least in the 19th century.” In 1934, the philosopher Paul Weiss called Peirce “the most original and versatile of American philosophers and America’s greatest logician” and Karl Popper considered him “one of the greatest philosophers of all times.” Clarence Irving Lewis (12.04.1883–3.02.1964) was an American logician, epistemologist, and moral philosopher, one of the founders of modern philosophical logic Lewis is best known in logic for his work exposing and criticizing the paradoxical features of the truth functional, material implication, as used in Russell and Whitehead’s Principia Mathematica, which declares, for instance, that any true consequent follows from any false antecedent Lewis proposed to replace that material implication with a strict implication which avoids that Lewis’ strict implication was not primitive, but defined as in terms of negation, conjunction, and a prefixed unary intensional modal operator ♦, where ♦A reads “A is possibly true.” Lewis then defined “A strictly implies B ” as ¬♦(A ∧ ¬B ) Lewis later devised the formal systems S to S of modal logic, adopting different principles for the modal operator ♦ and its dual (where A read as “A is necessarily true”), included in his 1932 book with Langford Symbolic Logic, thus laying the foundations of modern, formal modal logic